One Hat Cyber Team
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216.73.216.115
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194.44.31.54
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Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
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Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
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5.6.40
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Eksekusi
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~
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proc
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View File Name :
short.ps
%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: short.dvi %%Pages: 45 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o short.ps short.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.11.14:1337 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] 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restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 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y(n)m(um)m(b)s(er)33 b Fo(s)28 b Fn(2)g Fm(C)23 b Fq(\),)33 b(and)g(the)g(less)f(imp)s(ortan)m(t)f(function)383 1031 y Fo(R)q(e)p Fq(\()p Fo(W)14 b Fq(\)\()p Fo(s)p Fq(\))55 b(:=)1031 963 y Fo(R)q(e)p Fq(\()p Fo(s)p Fq(\)\()p Fo(R)q(e)p Fq(\()p Fo(s)p Fq(\))22 b Fn(\000)g Fq(1\))g Fn(\000)h Fo(I)8 b(m)2019 927 y Fe(2)2059 963 y Fq(\()p Fo(s)p Fq(\))p 1031 1007 1150 4 v 1361 1099 a Fn(j)28 b Fo(s)p Fq(\()p Fo(s)22 b Fn(\000)g Fq(1\))28 b Fn(j)1811 1070 y Fe(2)2190 1031 y Fo(;)45 b(s)27 b Fn(2)h Fm(C)46 b Fn(\000)23 b(f)p Fq(0)p Fo(;)17 b Fq(1)p Fn(g)p Fo(:)479 b Fq(\(1.9\))328 1306 y(In)49 b(particular)e(\(1.8\))h(and)h(\(1.9\))f (determine)g(t)m(w)m(o)i(p)s(olynomials)45 b(of)k(t)m(w)m(o)g(v)-5 b(ariables)328 1427 y Fo(I)8 b(;)17 b(R)28 b Fn(2)g Fm(Z)l Fq([)p Fo(U;)17 b(V)k Fq(])33 b(:)43 b(the)33 b Fr(fundamen)m(tal)378 1647 y Fo(I)8 b Fq(\()p Fo(U;)17 b(V)k Fq(\))27 b(:=)h Fo(V)21 b Fq(\(2)p Fo(U)26 b Fn(\000)15 b Fq(1\))28 b(=)f Fn(\000)p Fo(I)8 b(m)p Fq(\()p Fo(W)14 b Fq(\)\()p 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b Fq(:)27 b Fo(I)8 b Fq(\()p Fo(s)p Fq(\))28 b(=)f(0\))p Fn(g)p Fq(.)474 3370 y(According)i(to)g(the)h(form)e (\(1.10\))g(of)h Fo(I)37 b Fq(w)m(e)30 b(can)g(obtain)e(the)h(follo)m (wing)e(remark)-5 b(able)328 3490 y(implication)38 b(\(in)k(this)g(pap) s(er)g(this)g(implication)d(will)h(b)s(e)i(called)g(the)g Fr(Minor)49 b(Rie-)328 3610 y(mann)41 b(Hyp)s(othesis)35 b Fq(\(\(MRH\))h(in)f(short\),)i(according)e(to)g(its)g(formal)e(resem) m(blance)328 3731 y(\(lik)m(eness\))g(to)f(\(RH\):)603 3951 y(\()p Fo(M)10 b(R)q(H)e Fq(\))111 b Fo(I)8 b(f)38 b(I)8 b Fq(\()p Fo(s)p Fq(\))27 b(=)h(0)f Fo(and)h(I)8 b(m)p Fq(\()p Fo(s)p Fq(\))28 b Fn(6)p Fq(=)f(0)h Fo(then;)45 b(R)q(e)p Fq(\()p Fo(s)p Fq(\))28 b(=)f(1)p Fo(=)p Fq(2)p Fo(:)275 b Fq(\(1.13\))474 4171 y(In)26 b(this)f(pap)s(er)g(w)m(e)i (will)c(see)j(that)f(there)h(exists)g(a)f("dual")g(h)m(yp)s(othetical)f (statemen)m(t)328 4291 y(to)33 b(\(1.12\):)44 b(the)34 b(set)f(of)g Fr(all)k(zeros)h(of)h Fo(\020)h Fq(:)45 b Fo(\020)8 b Fq(\()p Fm(C)22 b Fq(\))29 b(=)g Fn(f)p Fo(s)f Fn(2)h Fm(C)53 b Fq(:)29 b Fo(\020)8 b Fq(\()p Fo(s)p Fq(\))27 b(=)i(0)p Fn(g)k Fq(is)g(con)m(tained)328 4411 y(in)f(the)h(set)g Fo(I)8 b Fq(\()p Fm(C)23 b Fq(\),)33 b(i.e.)1758 4532 y Fo(\020)8 b Fq(\()p Fm(C)23 b Fq(\))k Fn(\032)h Fo(I)8 b Fq(\()p Fm(C)24 b Fq(\))p Fo(:)1015 b Fq(\(1.14\))474 4706 y(The)38 b(inclusions)e(\(1.12\))g(and)i (\(1.14\))e(are)h(v)m(ery)h(surprising.)57 b(They)38 b(suggest)g(that)328 4827 y Fr(top)s(ological)23 b(information)e Fq(ab)s(out)h(isolated)f(p)s(oin)m(ts)h(of)g(the)h(meromorphic)d (function)328 4947 y Fo(\020)47 b Fq(is)39 b(written)g(de)h(facto)f(in) g(the)h Fr(algebraic)k Fm(R)9 b Fr(-v)-6 b(arieties)38 b Fo(I)8 b Fq(\()p Fm(C)24 b Fq(\))39 b(and)h Fo(I)8 b Fq(\()p Fm(C)23 b Fq(\))k Fn(\\)g Fo(R)q Fq(\()p Fm(C)d Fq(\),)1922 5214 y(4)p eop %%Page: 5 5 5 4 bop 328 631 a Fq(where)34 b Fo(I)40 b Fq(and)33 b Fo(R)g Fq(are)g(p)s(olynomials)c(from)j Fm(Z)-5 b Fq([)p Fo(U;)17 b(V)22 b Fq(])32 b(of)g(degree)i(2.)328 751 y(\(Ob)m(viously)k(only)f(zeros)i(and)g(p)s(oles)e(are)h(-)g(in)f(fact) h(-)f(in)m(teresting,)j(if)d(w)m(e)i(consider)f(a)328 872 y(meromorphic)31 b(function)h(from)f(the)i("top)s(ological")28 b(p)s(oin)m(t)k(of)g(view\).)474 992 y(The)27 b(aim)d(of)i(this)g(pap)s (er)g(is)f(th)m(us)i(to)f(explain)f(the)i(deep)s(er)g(sense)h(of)d(the) i Fr(Riemann)328 1112 y(h)m(yp)s(othesis)h Fq(as)g(some)g(relationship) e(b)s(et)m(w)m(een)k(the)e(cycles)h(:)41 b(the)28 b(holomorphic)e(man-) 328 1233 y(ifold)i Fo(\020)8 b Fq(\()p Fm(C)22 b Fq(\))30 b(and)g(the)g(algebraic)e(cycle)j Fo(I)8 b Fq(\()p Fm(C)23 b Fq(\))30 b(of)f(the)h(a\016ne)h(v)-5 b(ariet)m(y)30 b Fm(R)2904 1197 y Fe(2)2944 1233 y Fq(.)42 b(W)-8 b(e)30 b(are)g(going)328 1353 y(to)e(sho)m(w)i(that)e(a)g(formal)e(similarit)m (y)f(b)s(et)m(w)m(een)31 b(\(RH\))d(and)h(\(MRH\))f(is)g(not)h(only)f ("opti-)328 1474 y(cal,")i(but)g(there)i(exists)f(some)f Fr(unexp)s(ected)i Fq(\(and)f(hence)h Fr(deep)p Fq(\))f(relation)e(b)s (et)m(w)m(een)328 1594 y(the)e Fr(arithmetic)d Fq(of)i(the)h(p)s (olynomial)c Fo(I)34 b Fq(from)26 b Fm(Z)-5 b Fq([)p Fo(U;)17 b(V)k Fq(])27 b(o)m(v)m(er)h Fm(R)35 b Fq(and)26 b(the)h Fr(arithmetic)328 1714 y Fq(of)36 b Fo(\020)43 b Fq(o)m(v)m(er)37 b Fm(C)24 b Fq(\).)54 b(T)-8 b(o)36 b(b)s(e)h(more)e(exact,)j(when)f(thinking)e(o)m(v)m(er)j(these)f(ideas) f(it)f(b)s(ecomes)328 1835 y(more)h(and)h(more)f(apparen)m(t)h(to)g(us) g(that)g(the)g(Riemann)f(h)m(yp)s(othesis)i(on)e Fo(\020)44 b Fq(could)37 b(b)s(e)328 1955 y(reduced)d(to)e(the)h(formal)d (consequence)36 b(of)c(\(MRH\).)474 2076 y(In)46 b(this)f(pap)s(er)g(w) m(e)h(will)d(sho)m(w)j(-)f(what)g(w)m(as)h(also)e(quite)i(unexp)s (ected)h(for)e(us)g(-)328 2196 y(that)40 b(the)h(reduction)f(men)m (tioned)g(ab)s(o)m(v)m(e)h(can)g(b)s(e)f(obtained)g(b)m(y)h(using)f (only)g(simple)328 2316 y(calculations)34 b(of)h(in)m(tegrals)g(and)h (is)g(quite)g(elemen)m(tary)-8 b(.)53 b(\(A)m(t)36 b(this)g(momen)m(t)f (it)g(w)m(ould)328 2437 y(b)s(e)30 b(useful)h(to)e(recall)g(the)i(v)m (ery)g(short)g(F)-8 b(altings)28 b(pro)s(of)h([F)-8 b(a])30 b(of)f(the)i Fr(theorem)j(on)h(the)328 2557 y(\014nitness)48 b(of)h(the)f(n)m(um)m(b)s(er)g(of)h Fo(l)r Fr(-adic)f(c)m(haracters)h (of)f(represen)m(tations)h(of)328 2677 y(the)32 b(Galois)e(group)j(of)f (the)f(algebraic)g(closure)g(of)h(an)h(algebraic)e(\014eld)p Fq(,)d(whic)m(h)328 2798 y(w)m(as)38 b(the)g("psyc)m(hological)e (catalyst")i(for)f(our)g(pro)s(of.)58 b(Although)36 b(commonly)g(it)h (w)m(as)328 2918 y(considered)26 b(as)g(v)m(ery)h(hard)e(to)g(pro)m(v)m (e)i(\()e(e.g.)41 b(Serre)26 b([Se1])g(has)g(sho)m(w)m(ed)h(that)e(in)g (the)h(case)328 3039 y(of)40 b(elliptic)f(curv)m(es)k(it)d(follo)m(ws)g (from)f(the)j Fr(generalized)k(Riemann)h(h)m(yp)s(othesis)p Fq(\))328 3159 y(F)-8 b(altings)33 b(has)j(giv)m(en)f(a)g(pro)s(of)g ("on)g(one)g(page")g(using)g(the)h(old)e Fr(Hermite)k(theorem)328 3279 y Fq(from)44 b(the)j(Galois)c(theory)k(of)e(\014nite)g(extensions) i(and)f(the)h(tec)m(hnical)e Fr(Nak)-6 b(a)m(y)m(ama)328 3400 y(lemma)31 b Fq(\(it)h(is)g(also)f(necessary)k(to)e(apply)f(the)h (w)m(ell-kno)m(wn)g Fr(Cheb)s(otarev)38 b(densit)m(y)328 3520 y(theorem)p Fq(\(cf.e.g.)43 b([Se1]\)\)\).)474 3641 y(The)34 b(imaginary)c(part)i(of)g(t.f.e.)44 b(ob)m(viously)32 b(has)h(the)g(form)328 3910 y(\()p Fo(I)8 b(m)p Fq(\()p Fo(t:f)d(:e:)p Fq(\)\))29 b Fo(I)8 b(m)p Fq(\()p Fo(\020)1084 3869 y Fl(\003)1122 3910 y Fq(\()p Fo(s)p Fq(\)\))28 b(=)1413 3793 y Ff(Z)1496 3819 y Fe(+)p Fl(1)1459 3981 y Fe(1)1626 3910 y Fq(\()p Fo(x)1729 3835 y Fj(\()p Fg(u)p Fd(\000)p Fj(2\))p 1729 3854 164 4 v 1796 3895 a(2)1906 3910 y Fn(\000)p Fo(x)2038 3869 y Fl(\000)2103 3835 y Fj(\()p Fg(u)p Fj(+1\))p 2104 3854 162 4 v 2170 3895 a(2)2280 3910 y Fq(\))p Fo(sin)p Fq(\()2503 3843 y Fo(v)t(l)r(nx)p 2503 3887 196 4 v 2577 3978 a Fq(2)2709 3910 y(\))p Fo(\022)s Fq(\()p Fo(x)p Fq(\))p Fo(dx)p Fn(\000)3278 3843 y Fo(I)8 b Fq(\()p Fo(s)p Fq(\))p 3119 3887 489 4 v 3119 3978 a Fn(j)28 b Fo(s)p Fq(\()p Fo(s)22 b Fn(\000)h Fq(1\))k Fn(j)3569 3949 y Fe(2)3618 3910 y Fq(;)3317 4086 y(\(1.15\))328 4207 y Fo(s)h Fq(=)f Fo(u)22 b Fq(+)g Fo(iv)t Fq(.)474 4327 y(If)36 b(w)m(e)h(delete)f(the)h("p)s(erturbation")e Fo(\022)h Fq(=)e Fo(\022)s Fq(\()p Fo(x)p Fq(\))i(from)f(the)h(in)m (tegral)e(in)i Fo(I)8 b(m)p Fq(\()p Fo(t:f)d(:e:)p Fq(\))328 4447 y(and)46 b(c)m(hange)h(the)f(v)-5 b(ariables)45 b(according)g(to)h(the)g(substitution)g Fo(x)51 b Fq(=)f Fo(e)3066 4411 y Fe(2)p Fk(y)3143 4447 y Fq(,)f(then)e(w)m(e)328 4568 y(obtain)677 4701 y Ff(Z)760 4727 y Fe(+)p Fl(1)723 4889 y Fe(0)889 4818 y Fq(\()p Fo(e)972 4777 y Fk(uy)1077 4818 y Fn(\000)22 b Fo(e)1221 4777 y Fe(\(1)p Fl(\000)p Fk(u)p Fe(\))p Fk(y)1449 4818 y Fq(\))p Fo(sin)p Fq(\()p Fo(v)t(y)t Fq(\))p Fo(dy)85 b Fq(=)2314 4750 y Fo(I)8 b Fq(\()p Fo(s)p Fq(\))p 2156 4795 V 2156 4886 a Fn(j)27 b Fo(s)p Fq(\()p Fo(s)22 b Fn(\000)h Fq(1\))k Fn(j)2605 4857 y Fe(2)2682 4818 y Fq(;)44 b Fo(s)28 b Fq(=)f Fo(u)22 b Fq(+)g Fo(iv)t(:)1922 5214 y Fq(5)p eop %%Page: 6 6 6 5 bop 328 631 a Fq(The)33 b(ab)s(o)m(v)m(e)h(form)m(ula)c(\(or)i (rather)h Fr(rationalit)m(y)d Fq(of)i(the)h(in)m(tegral)e(as)i(a)f (function)g(of)g Fo(s)p Fq(\))328 751 y(w)m(as)h(the)g(direct)g (catalyst)f(of)g(our)h(pro)s(of.)474 872 y(Finally)-8 b(,)47 b(all)d(this)i(leads)g(to)g(the)h(follo)m(wing)c(main)i(result)h (of)g(this)f(pap)s(er)i(-)e(the)328 992 y Fr(Riemann)31 b(h)m(yp)s(othesis)g(functional)g(equation)c Fq(\(R.h.f.e)h(in)f (short\))g(with)g(the)h Fr(ra-)328 1112 y(tional)40 b(term)35 b Fo(I)44 b Fq(and)37 b(the)f Fr(action)41 b(of)h(the)g(Galois)e(group) d Fo(Gal)r Fq(\()p Fm(C)24 b Fo(=)p Fm(R)8 b Fq(\).)55 b(W)-8 b(e)36 b(ha)m(v)m(e)328 1233 y(just)41 b(men)m(tioned)g(that)g (suc)m(h)h(an)f(imp)s(ortan)m(t)e Fr(in)m(v)-6 b(arian)m(t)40 b Fq(of)g Fm(C)65 b Fq(as)41 b Fo(Gal)r Fq(\()p Fm(C)24 b Fo(=)p Fm(R)9 b Fq(\))41 b(m)m(ust)328 1353 y(ha)m(v)m(e)47 b(some)f(relation)f(with)g(\(RH\))h(and)h(it)e(is)g(really)g(so:)71 b(RH)46 b(is)g(an)g(immediately)328 1474 y(consequence)40 b(of)d(\(R.h.f.e.\))57 b(\(similarly)34 b(as)j(in)f(the)i(case)g(of)f (Matti)f(Pitk)-5 b(anen's)38 b(ana-)328 1594 y(lytic)e(pro)s(of)g(of)g (\(RH\))h([Pi])f(,)i(RH)f(is)f(an)h(immediate)d(consequence)40 b(of)c(the)h Fr(sp)s(ectral)328 1714 y(Hilb)s(ert-P)m(oly)m(a)25 b(conjecture)f Fq(pro)m(v)m(ed)i(b)m(y)f(him\).)39 b(Observ)m(e)26 b(that)e(in)g(fact)g(the)h("secret)328 1835 y(action")31 b(of)h Fo(Gal)r Fq(\()p Fm(C)24 b Fo(=)p Fm(R)9 b Fq(\))32 b(is)g(written)h(in)f(the)h(follo)m(wing)d(\()p Fo(R)q(hf)11 b(e)2635 1850 y Fk(G)2694 1835 y Fq(\).)474 1955 y(The)42 b(fundamen)m(tal)e(equation)g(men)m(tioned)h(ab)s(o)m(v)m(e)g(has)h (the)f(form:)58 b(there)42 b(exists)328 2076 y(suc)m(h)34 b(m)m(ultiplicators)29 b Fo(F)1232 2091 y Fe(1)1272 2076 y Fq(\()p Fo(s)p Fq(\))p Fo(;)17 b(F)1501 2091 y Fe(2)1540 2076 y Fq(\()p Fo(s)p Fq(\))32 b(and)h Fo(f)1932 2091 y Fe(1)1971 2076 y Fq(\()p Fo(s)p Fq(\))p Fo(;)17 b(f)2185 2091 y Fe(2)2224 2076 y Fq(\()p Fo(s)p Fq(\))33 b(with)f Fo(f)2649 2091 y Fe(1)2710 2076 y Fn(\000)23 b Fo(f)2858 2091 y Fe(2)2925 2076 y Fn(6)p Fq(=)28 b(0)k(that)328 2323 y(\()p Fo(R)q(:h:f)5 b(:e)676 2338 y Fk(G)736 2323 y Fq(\))55 b Fo(I)8 b(m)p Fq(\()1118 2240 y Ff(X)1003 2441 y Fk(g)r Fl(2)p Fk(M)f(od)p Fe(\()p Fm(C)25 b Fe(\))1352 2323 y Fq(\()p Fo(F)1453 2338 y Fe(1)1493 2323 y Fo(\020)8 b Fq(\)\()p Fo(g)t Fq(\()p Fo(s)p Fq(\)\))26 b(+)h(\()p Fo(F)2061 2338 y Fe(2)2101 2323 y Fo(\020)8 b Fq(\)\()p Fo(c)p Fq(\()p Fo(g)t Fq(\()p Fo(s)p Fq(\)\)\))53 b(=)2751 2256 y(\()p Fo(f)2837 2271 y Fe(1)2876 2256 y Fq(\()p Fo(s)p Fq(\))22 b Fn(\000)h Fo(f)3168 2271 y Fe(2)3208 2256 y Fq(\()p Fo(s)p Fq(\)\))p 2751 2300 617 4 v 2843 2391 a Fn(j)p Fo(s)p Fq(\()p Fo(s)e Fn(\000)i Fq(1\))p Fn(j)3237 2363 y Fe(2)3377 2323 y Fo(I)8 b Fq(\()p Fo(s)p Fq(\))p Fo(:)3317 2549 y Fq(\(1.16\))474 2669 y(\(Assume)36 b(that)e(for)g(some)h Fo(s)p Fq(,)g(with)f Fo(I)8 b(m)p Fq(\()p Fo(s)p Fq(\))31 b Fn(6)p Fq(=)g(0)k(and)g Fo(R)q(e)p Fq(\()p Fo(s)p Fq(\))c Fn(6)p Fq(=)g(1)p Fo(=)p Fq(2,)k(is)f Fo(\020)8 b Fq(\()p Fo(s)p Fq(\))30 b(=)h(0.)328 2790 y(Then)38 b(also)e Fo(\020)8 b Fq(\()p Fo(c)p Fq(\()p Fo(s)p Fq(\)\))33 b(=)i(0)h(and)h(the)g(left-hand)f(side)h(of)f(\()p Fo(R)q(:h:f)5 b(:e:)2782 2805 y Fk(G)2842 2790 y Fq(\))37 b(is)f(zero)h(whereas)328 2910 y(the)c(righ)m(t-hand)f(side)g Fr(is)37 b(not)p Fq(.)43 b(This)33 b(pro)m(v)m(es)h(\(RH\)\).)474 3030 y(\()p Fo(R)q(:h:f)5 b(:e:)849 3045 y Fk(G)909 3030 y Fq(\))35 b(is)e(the)i(cro)m(wning)f(of)g(the)h Fr(algebraic)k(t)m(yp) s(e)34 b Fq(conjecture,)i(whic)m(h)f(w)m(e)328 3151 y(deriv)m(ed)k (during)e(our)h(\014rst)h(visit)e(in)g(U.S.A.)i(ten)f(y)m(ears)h(ago)f (and)g(in)m(tro)s(ductory)g(dis-)328 3271 y(cused)i(with)f(Chris)f (Burdzy)-8 b(.)64 b(That)39 b(equation)f(th)m(us)i(pro)m(v)m(es)h(the)e Fr(main)44 b(algebraic)328 3392 y(conjecture)33 b Fq(p)s(osed)g(b)m(y)g (us)g(ten)g(y)m(ears)h(ago)e(:)1347 3584 y(\()p Fo(M)10 b(AC)d Fq(\))84 b(\()p Fo(S)6 b(R)q(H)i Fq(\))27 b(=)-17 b Fn(\))27 b Fq(\()p Fo(R)q(H)8 b Fq(\))p Fo(:)770 b Fq(\(1.17\))328 3777 y(In)33 b([M)s(\030)-46 b(a2])32 b(and)h([M)s(\030)-46 b(a3])33 b(w)m(e)g(pro)m(v)m(ed)h(\(MA)m(C\))g (in)e(a)g(quite)h(di\013eren)m(t)g(manner.)43 b(In)33 b([M)s(\030)-46 b(a2])328 3897 y(w)m(e)35 b(ha)m(v)m(e)h(pro)m(v)m(ed)g (that)e(\(MA)m(C\))h(is)f(an)g(immediate)d(consequence)38 b(of)c(the)g Fr(existence)328 4017 y Fq(of)26 b(the)i(so)f(called)e Fr(Ho)s(dge)31 b(measure)g Fo(H)1841 4032 y Fe(2)1907 4017 y Fq(\(i.e.)41 b Fo(\033)t Fq(-additiv)m(e)26 b(p)s(ositiv)m(e)g (measure)i(on)e(the)328 4138 y(Banac)m(h)31 b(algebra)e Fo(C)1083 4153 y Fe(0)1122 4138 y Fq([0)p Fo(;)17 b Fq(1])1318 4102 y Fe(2)1357 4138 y Fq(,)31 b(where)h Fo(C)1765 4153 y Fe(0)1804 4138 y Fq([0)p Fo(;)17 b Fq(1])30 b(is)g(the)h(Banac)m(h)f (algebra)g(of)g(all)e(con)m(tin)m(u-)328 4258 y(ous)j(functions)g Fo(f)41 b Fq(on)30 b([0)p Fo(;)17 b Fq(1])31 b(with)f Fo(f)11 b Fq(\(0\))27 b(=)g(1\),)k(whic)m(h)g(giv)m(es)g(the)g(follo)m (wing)d Fr(Abstract)328 4379 y(Ho)s(dge)38 b(Decomp)s(osition)30 b Fq(\()p Fo(AH)8 b(D)1721 4394 y Fe(2)1760 4379 y Fq(\))33 b(of)f(the)h(2-dimensional)c Fr(Green)38 b(function)p Fq(:)656 4601 y(\()p Fo(AH)8 b(D)937 4616 y Fe(2)976 4601 y Fq(\))111 b Fn(j)27 b Fo(s)h Fn(j)1282 4560 y Fl(\000)p Fe(2)1404 4601 y Fq(=)1535 4484 y Ff(Z)1581 4673 y Fk(C)1631 4682 y Fj(0)1666 4673 y Fe([0)p Fk(;)p Fe(1])p Fl(\002)p Fk(C)1901 4682 y Fj(0)1935 4673 y Fe([0)p Fk(;)p Fe(1])2085 4601 y Fo(e)2130 4560 y Fe([)p Fk(s;c)p Fe(])2257 4601 y Fo(dH)2389 4616 y Fe(2)2428 4601 y Fq(\()p Fo(c)p Fq(\))p Fo(;)71 b Fq(;)17 b Fo(s)28 b Fn(2)g Fm(C)2922 4560 y Fl(\003)2961 4601 y Fo(:)329 b Fq(\(1.18\))328 4844 y(Here)51 b([)p Fo(:;)17 b(:)p Fq(])51 b(is)f(some)g(\(rather)g (complicated\))f(bilinear)f(form)i(on)g Fo(C)7 b Fq([0)p Fo(;)17 b Fq(1])34 b Fn(\002)g Fo(C)7 b Fq([0)p Fo(;)17 b Fq(1])328 4965 y(\(the)41 b(so)f(called)g(Mik)m(olasc)m(h-Ko)s(c)m(h) g(error\).)67 b(W)-8 b(e)41 b(can)g(succin)m(tly)f(sa)m(y)i(that)e (existence)1922 5214 y(6)p eop %%Page: 7 7 7 6 bop 328 631 a Fq(of)50 b Fo(AH)8 b(D)700 646 y Fe(2)789 631 y Fq(means)51 b(that)f(the)h(2-dimensional)d(Green)i(function)g(is) g(a)h Fr(functional)328 751 y(Laplace)38 b(transform)32 b Fq(of)g Fo(H)1440 766 y Fe(2)1480 751 y Fq(,)g(i.e.)43 b Fn(j)28 b Fo(s)f Fn(j)1864 715 y Fl(\000)p Fe(2)1958 751 y Fq(=)2097 726 y(^)2062 751 y Fo(H)2143 766 y Fe(2)2182 751 y Fq(\()p Fo(s)p Fq(\))32 b(\(cf.[M)s(\030)-46 b(a2,Th.1]\).)474 872 y(\()p Fo(AH)8 b(D)755 887 y Fe(2)794 872 y Fq(\))22 b(implies)d(in)i(an)h(elemen)m(tary)g(w)m(a)m(y)h(the)f(\()p Fo(R)q(:h:f)5 b(:e:)2609 887 y Fe(2)2650 872 y Fq(\))21 b(of)h(the)g(form)e(\(cf.[M)s(\030)-46 b(a2,)328 992 y(Th.2]\))1195 1167 y(\()p Fo(R)q(hf)11 b(e)1468 1182 y Fe(2)1507 1167 y Fq(\))133 b Fn(\000)1833 1100 y Fo(I)8 b(m)p Fq(\()p Fo(\020)2058 1064 y Fl(\003)2097 1100 y Fq(\()p Fo(s)p Fq(\)\))p 1788 1144 515 4 v 1788 1236 a Fo(T)14 b(r)1903 1251 y Fe(2)1942 1236 y Fq(\()p Fo(M)2074 1251 y Fk(G)2133 1236 y Fo(M)2227 1251 y Fk(s)2265 1236 y Fq(\))2368 1167 y(=)55 b Fo(I)8 b Fq(\()p Fo(s)p Fq(\))p Fo(;)618 b Fq(\(1.19\))328 1394 y(where)34 b(the)f("analytic)e(Riemann) g(trace")i Fo(T)14 b(r)2019 1409 y Fe(2)2058 1394 y Fq(\()p Fo(M)2190 1409 y Fk(G)2249 1394 y Fo(M)2343 1409 y Fk(s)2380 1394 y Fq(\))33 b(is)f(giv)m(en)h(b)m(y)g(the)g(form)m(ula)383 1673 y Fo(T)14 b(r)498 1688 y Fe(2)537 1673 y Fq(\()p Fo(M)669 1688 y Fk(G)729 1673 y Fo(M)823 1688 y Fk(s)860 1673 y Fq(\))55 b(:)1040 1565 y Fe(+)p Fl(1)1042 1590 y Ff(X)1036 1771 y Fk(n)p Fe(=0)1185 1565 y(+)p Fl(1)1188 1590 y Ff(X)1187 1772 y Fk(j)t Fe(=0)1337 1606 y Fq(\()p Fn(\000)p Fo(\031)t(n)1569 1569 y Fe(2)1609 1606 y Fq(\))1647 1569 y Fk(j)p 1337 1650 347 4 v 1474 1741 a Fo(j)6 b Fq(!)1716 1673 y Fn(\001)2104 1606 y Fq(\(4)p Fo(j)28 b Fq(+)22 b(1\))p 1776 1650 996 4 v 1776 1741 a Fn(j)27 b Fq(\()p Fo(s)22 b Fn(\000)h Fq(2)p Fo(j)6 b Fq(\)\()p Fo(s)21 b Fq(+)h(2)p Fo(j)28 b Fn(\000)23 b Fq(1\))k Fn(j)2732 1713 y Fe(2)2781 1673 y Fo(;)509 b Fq(\(1.20\))328 1966 y(and)48 b(0)k Fo(<)h(T)14 b(r)878 1981 y Fk(an)962 1966 y Fq(\()p Fo(M)1094 1981 y Fk(G)1154 1966 y Fo(M)1248 1981 y Fk(s)1285 1966 y Fq(\))53 b Fo(<)g Fq(+)p Fn(1)p Fq(.)88 b(\(It)47 b(is)g(v)m(ery)i(con)m(v)m(enien)m(t)g(to)e(c)m(ho)s (ose)i(a)e Fr(non-)328 2087 y(canonical)37 b Fq(scenario:)53 b(0)1288 2051 y Fe(0)1363 2087 y Fq(=)36 b(1\).)57 b(Ho)m(w)m(ev)m(er) 40 b(an)d(anon)m(ymous)h(referee)h(from)d(Crelle's)328 2207 y(Journal)g(in)g(his)h(rep)s(ort)g([CJR])g(on)g([M)s(\030)-46 b(a2])37 b(dated)h(F)-8 b(ebruary)37 b(26,)h(\(1999\))e(disco)m(v)m (ered)328 2327 y(n)m(umerical)26 b(coun)m(terexamples)j(to)e(\()p Fo(R)q(:h:f)5 b(:e:)1992 2342 y Fe(2)2032 2327 y Fq(\).)42 b(He)28 b(denoted)h Fo(h)p Fq(\()p Fo(s)p Fq(\))e(=)h Fo(\020)3007 2291 y Fl(\003)3045 2327 y Fq(\()p Fo(s)p Fq(\))p Fo(=)g Fn(j)f Fo(\020)3350 2291 y Fl(\003)3389 2327 y Fq(\()p Fo(s)p Fq(\))g Fn(j)328 2448 y Fq(and)33 b(obtained:)827 2658 y Fo(h)p Fq(\(0)p Fo(:)p Fq(6)22 b(+)g(15)p Fo(:)p Fq(4)p Fo(i)p Fq(\))55 b(=)g(0)p Fo(:)p Fq(9999989257)p Fo(:::)20 b Fq(+)i Fo(i)p Fq(0)p Fo(:)p Fq(0014657814)p Fo(:::;)865 2868 y(h)p Fq(\(0)p Fo(:)p Fq(9)f(+)h(20)p Fo(i)p Fq(\))56 b(=)f(0)p Fo(:)p Fq(7882729297)p Fo(:::)20 b Fq(+)i Fo(i)p Fq(0)p Fo(:)p Fq(6153257577)p Fo(::::)328 3039 y Fq(The)27 b(reason)g(and)g(explanation)e(of)h(this)g (con)m(tradiction)g(is)g(that)g(the)h(pro)s(of)e(of)h(\()p Fo(R)q(hf)11 b(e)3488 3054 y Fe(2)3528 3039 y Fq(\))328 3159 y(is)40 b(strongly)g(based)h(on)g(the)g(existence)h(of)d(the)i(Ho) s(dge)g(measure)f Fo(H)2915 3174 y Fe(2)2955 3159 y Fq(.)67 b(Ho)m(w)m(ev)m(er,)45 b(as)328 3279 y(shortly)33 b(after)f(w)m(as)i (observ)m(ed)g(b)m(y)g(M.Bo)8 b(_)-35 b(zejk)m(o,)34 b Fr(on)k(the)g(ground)g(of)g(the)f(classical)328 3400 y(logic)p Fq(,)k(the)g(measure)f Fo(H)1266 3415 y Fe(2)1346 3400 y Fr(cannot)47 b(exist)p Fq(,)42 b(since)2302 3375 y(^)2267 3400 y Fo(H)2348 3415 y Fe(2)2387 3400 y Fq(\()p Fo(s)p Fq(\))e(is)g Fr(p)s(ositiv)m(e)45 b(de\014nite)c Fq(in)328 3520 y(the)d(Laplace-Haenk)m(el)g(sense)i(on)e Fm(C)1699 3484 y Fl(\003)1777 3520 y Fq(whereas)h(the)g(Green)f (function)g Fn(j)e Fo(:)h Fn(j)3162 3484 y Fl(\000)p Fe(2)3294 3520 y Fq(is)h(not.)328 3641 y(Herceforth)33 b(this)f(is)h(refered)g(to)f(as)h(the)g Fr(Bo)9 b(_)-40 b(zejk)m(o)37 b(parado)m(x)p Fq(.)474 3761 y(Ho)m(w)m(ev)m(er,)46 b(if)40 b(w)m(e)i(reject)g(the)f Fr(T)-9 b(ertium)45 b(non)j(Datur)41 b Fq(\(TnD)g(in)g(short\),)i(\(then)328 3881 y(also)33 b(ad)h(absurdum)g(metho)s(d)f(of)h(pro)s(of)7 b(\),)33 b(i.e.)47 b(w)m(e)35 b(agree)f(to)f(w)m(ork)i(on)f(the)g (ground)g(of)328 4002 y Fr(Brou)m(w)m(er)41 b(in)m(tuicionistic)c (logic)p Fq(,)f(then)h Fo(H)2049 4017 y Fe(2)2125 4002 y Fq(exists.)55 b(In)37 b(the)g(construction)f(of)g Fo(H)3499 4017 y Fe(2)3539 4002 y Fq(,)328 4122 y(in)d(a)h(secret)h(w)m(a)m(y)g (app)s(ears)f(famous)g(diagonal)d(t)m(yp)s(e)k(construction)f(from)f (the)h Fr(Go)s(del)328 4242 y(theorem)p Fq(.)52 b(Constructions)36 b(ha)m(v)m(e)h(b)s(een)f(kno)m(wn)h(for)e(a)g(h)m(undred)j(y)m(ears)e (whic)m(h)g(exist)328 4363 y(only)e(on)h(the)h(lev)m(el)e(of)h(Brou)m (w)m(er)h(logic)d(\(or)h(the)i(set)f(theory)h(is)e(con)m(tradictory)h (!\):)48 b(for)328 4483 y(example)40 b Fr(diagonal)45 b(Brou)m(w)m(er)h(set)40 b Fq(of)g(in)m(tegers)g Fo(D)s Fm(N)52 b Fq(\(for)40 b(whic)m(h)h(the)f(statemen)m(t)328 4604 y(\()p Fo(n)28 b Fn(2)g Fo(D)s Fm(N)12 b Fq(\))s Fn(_)s Fq(\()p Fo(notinD)s Fm(N)h Fq(\)\))23 b(or)g(the)g Fr(Sp)s(ec)m(k)m(er)k(sequence)d Fn(f)p Fo(s)2546 4619 y Fk(n)2593 4604 y Fn(g)f Fq(\(monotonic,)g(b)s(ounded)328 4724 y(and)33 b(div)m(ergen)m(t\))g(\(cf.[Br],[Sp],[Ric],[ML]\).)474 4844 y(Since)43 b(w)m(e)h(could)e(not)g(c)m(hec)m(k)j(directly)d(the)h (correctness)h(of)f(the)g(computer)f(cal-)328 4965 y(culations)36 b(and)i(did)e(not)i(en)m(tirely)f(b)s(eliev)m(e)g(them,)h(w)m(e)h(sen)m (t)f(the)g(pap)s(er)g(to)f(the)h(Acta)1922 5214 y(7)p eop %%Page: 8 8 8 7 bop 328 631 a Fq(Arithmetica)27 b(Journal.)42 b(An)29 b(anon)m(ymous)h(referee)g(from)e(AA)i(\(cf.[AAR]\))f(wrote)h(:"w)m(e) 328 751 y(see)38 b(therefore)g(that)f(the)h(claimed)d("Riemann)h(h)m (yp)s(othesis)i(equation")f(\()p Fo(R)q(hf)11 b(e)3332 766 y Fe(2)3371 751 y Fq(\))38 b(to-)328 872 y(gether)30 b(with)g(0)d Fo(<)h(T)14 b(r)1138 887 y Fe(2)1177 872 y Fq(\()p Fo(M)1309 887 y Fk(G)1369 872 y Fo(M)1463 887 y Fk(s)1500 872 y Fq(\))27 b Fo(<)h Fn(1)h Fq(implies)f(b)s(oth)i(the)g (Riemann)f(h)m(yp)s(othesis)i(and)328 992 y(its)j(negation.)49 b(This)35 b(sho)m(ws)h(ob)m(viously)e(that)h(at)f(least)g(one)h(of)f (the)h(form)m(ulae)f(cannot)328 1112 y(b)s(e)e(true".)44 b(W)-8 b(e)33 b(add:)43 b("ob)m(viously")32 b(on)g(the)h(ground)f(of)g Fr(classical)k(logic)p Fq(.)41 b(Moreo)m(v)m(er,)328 1233 y(w)m(e)36 b(see)f(that)g(on)f(the)h(ground)f(of)h(Brou)m(w)m(er)g (logic)e(,)i(\(RH\))f(is)g(exactly)h(the)g(statemen)m(t)328 1353 y(whic)m(h)e Fr(violates)e Fq(\(TnD\))i(itself.)474 1474 y(The)j(series)f(of)e Fr(de)40 b(Branges')g(pap)s(ers)c Fq(dev)m(oted)g(to)e(the)h(pro)s(of)e(of)h(\(RH\):)g([dB1-)328 1594 y(dB5])25 b(has)g(b)s(een)g(kno)m(wn)h(since)f(1994.)40 b(His)25 b("pro)s(of")e(w)m(as)j(in)e(man)m(y)g(p)s(oin)m(ts)h(v)m(ery) h(similar)328 1714 y(to)f(our)h(pro)s(of)f(in)g([M)s(\030)-46 b(a2]:)40 b(he)27 b(also)e(uses)i(some)f(in)m(tegral)e(transforms)h (\(in)g(the)i(adic)e(case\))328 1835 y(and)37 b(sho)m(ws)h(that)f (\(RH\))f(is)h(a)f(consequence)k(of)c(the)h(p)s(ositivit)m(y)f(of)g (his-)g(so-called)f Fr(de)328 1955 y(Branges)47 b(trace)40 b Fq(:)p Fo(R)q(e)h(<)g(F)14 b Fq(\()p Fo(z)t Fq(\))p Fo(;)j(F)d Fq(\()p Fo(z)32 b Fq(+)c Fo(i)p Fq(\))41 b Fo(>)2088 1971 y Fl(H)p Fe(\()p Fk(E)t Fe(\))2262 1955 y Fn(\025)h Fq(0.)67 b(More)40 b(exactly)h(:)59 b(let)40 b Fo(E)6 b Fq(\()p Fo(z)t Fq(\))328 2076 y(b)s(e)47 b(an)g(en)m(tire)g (function)g(sat)m(ysfying)h Fn(j)j Fo(E)6 b Fq(\()p Fo(c)p Fq(\()p Fo(z)t Fq(\)\))53 b Fn(j)p Fo(<)p Fn(j)f Fo(E)6 b Fq(\()p Fo(z)t Fq(\))53 b Fn(j)46 b Fq(for)h Fo(z)52 b Fq(in)46 b(the)h(upp)s(er)328 2196 y(half-plane.)54 b(A)37 b(Hilb)s(ert)e(space)j(of)e(en)m(tire)h(functions)f Fn(H)q Fq(\()p Fo(E)6 b Fq(\))37 b(is)f(the)h(set)h(of)e(all)f(en)m (tire)328 2316 y(functions)29 b Fo(F)14 b Fq(\()p Fo(z)t Fq(\))29 b(suc)m(h)i(that)e Fo(F)14 b Fq(\()p Fo(z)t Fq(\))p Fo(=E)6 b Fq(\()p Fo(z)t Fq(\))29 b(is)g(square)h(in)m (tegrable)e(on)h(the)h(real)e(axis)h(and)328 2437 y(suc)m(h)34 b(that)1396 2557 y Fn(j)28 b Fo(F)14 b Fq(\()p Fo(z)t Fq(\))28 b Fn(j)1710 2516 y Fe(2)1749 2557 y Fn(\024jj)f Fo(F)42 b Fn(jj)2070 2516 y Fe(2)2070 2582 y Fl(H)p Fe(\()p Fk(E)t Fe(\))2271 2557 y Fo(K)7 b Fq(\()p Fo(z)t(;)17 b(z)t Fq(\))738 b(\(1.21\))328 2724 y(for)32 b(all)e(complex)j Fo(z)t Fq(,)g(where)h(the)f(inner)f(pro)s(duct)h(of)f(the)h(space)h(is) e(giv)m(en)g(b)m(y)1036 2977 y Fo(<)27 b(F)14 b Fq(\()p Fo(z)t Fq(\))p Fo(;)j(G)p Fq(\()p Fo(z)t Fq(\))28 b Fo(>)1691 2992 y Fl(H)p Fe(\()p Fk(E)t 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Fq(\()p Fo(z)t Fq(\))p Fo(;)j(K)7 b Fq(\()p Fo(w)s(;)17 b(z)t Fq(\))28 b Fo(>)2486 3972 y Fl(H)p Fe(\()p Fk(E)t Fe(\))328 4160 y Fq(holds)43 b(for)g(ev)m(ery)j(complex)d Fo(w)j Fq(and)d(for)g(ev)m(ery)j(elemen)m (t)d Fo(F)60 b Fn(2)47 b(H)q Fq(\()p Fo(E)6 b Fq(\).)76 b(The)45 b(ab)s(o)m(v)m(e)328 4280 y(iden)m(tit)m(y)29 b(is)g(obtained)g(b)m(y)h(using)g(Cauc)m(h)m(y's)i(in)m(tegration)27 b(form)m(ula)h(in)h(the)g(upp)s(er)h(half-)328 4400 y(plane)38 b(\(cf.[dB1]\),)j(and)d(the)h(condition)e(\(1.2\))h(is)g(made)g(so)h (that)g(Cauc)m(h)m(y's)i(form)m(ula)328 4521 y(applies)32 b(to)g(all)f(functions)i(in)f(the)h(space)h Fn(H)q Fq(\()p Fo(E)6 b Fq(\).)44 b(In)33 b(other)g(w)m(ords)g Fn(H)q Fq(\()p Fo(E)6 b Fq(\))33 b(is)f(a)h(RKHS)328 4641 y(\(Repro)s(ducing) 42 b(Kernel)h(Hilb)s(ert)e(Space\).)75 b(According)42 b(to)h(de)g(Branges)g(results)g(for)328 4762 y Fo(E)6 b Fq(\()p Fo(z)t Fq(\))33 b(and)g(some)f(additional)e("tec)m(hnical)i (conditions"\(cf.[dB2-dB5]\),)f(if)1290 4965 y Fo(R)q(e)d(<)g(F)14 b Fq(\()p Fo(z)t Fq(\))p Fo(;)j(F)d Fq(\()p Fo(z)26 b Fq(+)c Fo(i)p Fq(\))28 b Fo(>)2247 4980 y Fl(H)p Fe(\()p Fk(E)t Fe(\))2505 4965 y Fn(\025)56 b Fq(0)630 b(\(1.22\))1922 5214 y(8)p eop %%Page: 9 9 9 8 bop 328 631 a Fq(for)41 b(ev)m(ery)j(elemen)m(t)e Fo(F)14 b Fq(\()p Fo(z)t Fq(\))44 b Fn(2)g(H)q Fq(\()p Fo(E)6 b Fq(\))42 b(with)g Fo(F)14 b Fq(\()p Fo(z)33 b Fq(+)28 b Fo(i)p Fq(\))44 b Fn(2)g(H)q Fq(\()p Fo(E)6 b Fq(\),)44 b(then)f(the)f(zeros)h(of)328 751 y Fo(E)6 b Fq(\()p Fo(z)t Fq(\))36 b(lie)d(on)i(the)h(line)d Fo(I)8 b(m)p Fq(\()p Fo(z)t Fq(\))33 b(=)f Fn(\000)p Fq(1)p Fo(=)p Fq(2)i(\(i.e.)51 b(satis\014es)35 b(\(RH\)\))g(and)g(moreo)m(v)m (er)h Fo(R)q(e)c(<)328 872 y(c)p Fq(\()p Fo(E)6 b Fq(\))524 836 y Fl(0)547 872 y Fq(\()p Fo(w)s Fq(\))p Fo(E)g Fq(\()p Fo(w)31 b Fq(+)f Fo(i)p Fq(\))p Fo(=)p Fq(2)p Fo(\031)t(i)46 b(>)93 b Fq(0)46 b Fn(\025)e Fq(when)g Fo(w)i Fq(is)d(a)g(zero)h(of)f Fo(E)6 b Fq(\()p Fo(z)t Fq(\))44 b(\(i.e.)76 b(b)s(elongs)43 b(to)328 992 y(the)49 b(algebraic)d(v)-5 b(ariete)48 b Fo(E)6 b Fq(\()p Fm(C)24 b Fq(\)\).)90 b(Finally)46 b(he)j(applied)e(the)h(theory)h(in)f(the)h(case)g(:)328 1112 y Fo(E)6 b Fq(\()p Fo(z)t Fq(\))28 b(=)g(\()p Fo(iz)f Fn(\000)c Fq(1\))p Fo(iz)t(\020)1126 1076 y Fl(\003)1165 1112 y Fq(\(1)f Fn(\000)g Fo(iz)t Fq(\))34 b(to)e(obtain)g(\(RH\).)474 1233 y(Ho)m(w)m(ev)m(er)44 b(Conrey)f(and)e(Li)g(in)g([CL])h(sho)m(w)m (ed)h(the)f(falsit)m(y)f(of)g(de)h(Branges's)g(ap-)328 1353 y(proac)m(h)37 b(to)f(\(RH\).)g(They)i(used)g(MA)-8 b(THEMA)g(TICA)39 b(and)e(n)m(umerically)d(pro)m(v)m(ed)k(that)328 1474 y(de)g(Branges')h Fr(p)s(ositivit)m(y)i(conditions)p Fq(,)d(whic)m(h)g(imply)e(the)j(generalized)e(Riemann)328 1594 y(h)m(yp)s(othesis,)53 b(are)c(not)f(satis\014ed)h(b)m(y)g (de\014ning)f(functionsof)g(the)h(repro)s(ducing)f(k)m(er-)328 1714 y(nel)38 b(Hilb)s(ert)e(spaces)k(asso)s(ciated)e(with)f(the)i (Riemann)d(zeta)j(function)e Fo(\020)8 b Fq(\()p Fo(s)p Fq(\))37 b(and)h(the)328 1835 y(Diric)m(hlet)30 b Fo(L)p Fq(-function)i Fo(L)p Fq(\()p Fo(s;)17 b(\037)1466 1850 y Fe(4)1506 1835 y Fq(\))32 b(\(cf.[CL,\(3.2\),\(3.4\)and)h(Section)f (3.2]\).)474 1955 y(My)24 b(explanation)e(of)g(this)h(phenomen)h(is)e (the)i(same)f(as)g(ab)s(o)m(v)m(e)h(:)38 b(the)24 b(adic)e(construc-) 328 2076 y(tion)35 b(of)h(de)g(Branges)h Fr(violates)e Fq(\(TnD\))h(and)g(exists)h(only)f(on)g(the)g(lev)m(el)g(of)f(Brou)m(w) m(er)328 2196 y(logic!)474 2316 y(In)40 b([M)s(\030)-46 b(a3])40 b(w)m(e)g(pro)m(v)m(ed)h(the)f(existence)h(of)e(a)g(measure)h (quite)f(di\013eren)m(t)h(from)e(the)328 2437 y Fo(H)409 2452 y Fe(2)482 2437 y Fr(Ho)s(dge)h(measure)c Fo(H)1355 2452 y Fe(0)1394 2437 y Fq(.)47 b(The)35 b(measure)f Fo(H)2132 2452 y Fe(0)2206 2437 y Fq(also)f Fr(exists)38 b(only)33 b Fq(on)h(the)h(lev)m(el)e(of)328 2557 y(Brou)m(w)m(er)24 b(logic)d(\(according)h(to)g(the)h Fr(Hardy-Littlew)m(o)s(o)s(d)i (theorem)p Fq(,)f(whic)m(h)f(asserts)328 2677 y(that)k(on)h(the)g (critical)d(line)h Fo(R)q(e)p 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b(the)h(algebraic)e(v)-5 b(arieties)23 b(o)m(v)m(er)i(\014nite)e(\014elds)h(\(W)-8 b(eil)22 b(conjectures)j(-)f(Grothendiec)m(k-)328 4842 y(Deligne)d(theorems\))i (-)f(is)g(a)g(consequence)k(of)c(the)h Fr(Lefsc)m(hetz)j(\014xed)h(p)s (oin)m(ts)f(form)m(ula)328 4962 y Fq(in)40 b(W)-8 b(eil)38 b(cohomologies.)65 b(Hence)42 b(the)e(surprising)g(role)g(of)g(the)g (general)g(\014xed)i(p)s(oin)m(t)1922 5214 y(9)p eop %%Page: 10 10 10 9 bop 328 631 a Fq(theory)28 b(\(cf.e.g.)42 b([DG]\))27 b(in)g(the)h(Riemann)e(problem\).)41 b(In)28 b([M)s(\030)-46 b(a3])28 b(w)m(e)g(also)f(pro)m(v)m(ed)i(that)328 751 y(\()p Fo(AH)8 b(D)609 766 y Fe(0)648 751 y Fq(\))30 b(implies)d(the)j(existence)h(of)e(a)h(functional)e(equation)h(of)h (the)g(follo)m(wing)d(form:)328 872 y(there)33 b(exists)h(suc)m(h)g(a)e (sequence)j(of)d(symmetric)g(sub-Dirac)g(functions)g Fo(D)3086 836 y Fk(n)3083 896 y Fe(0)3133 872 y Fq(\()p Fo(t)p Fq(\))g(that)399 1147 y(\()p Fo(R)q(:h:f)5 b(:e:)774 1162 y Fe(0)815 1147 y Fq(\))68 b(lim)897 1196 y Fk(n)p Fl(!1)1097 1030 y Ff(Z)1144 1218 y Fe(\012)1195 1227 y Fj(0)1250 1147 y Fo(I)8 b(m)p Fq(\(\(\000\()p Fo(!)25 b Fq(+)d(\()p Fo(D)1867 1106 y Fk(n)1864 1171 y Fe(0)1936 1147 y Fq(+)2057 1122 y(^)2034 1147 y Fo(D)2118 1106 y Fk(n)2115 1171 y Fe(0)2165 1147 y Fq(\)\))p Fo(\020)8 b Fq(\)\()p Fo(s)p Fq(\)\))p Fo(dH)2622 1162 y Fe(0)2659 1147 y Fq(\()p Fo(!)t Fq(\))55 b(=)3001 1079 y Fo(H)3082 1094 y Fe(0)3121 1079 y Fq(\(\012)3229 1094 y Fe(0)3269 1079 y Fq(\))p Fo(I)8 b Fq(\()p Fo(s)p Fq(\))p 2996 1123 489 4 v 2996 1215 a Fn(j)27 b Fo(s)p Fq(\()p Fo(s)22 b Fn(\000)h Fq(1\))k Fn(j)3445 1186 y Fe(2)3317 1323 y Fq(\(1.24\))328 1443 y(,)33 b(whic)m(h)g(ob)m(viously)f(immediately)e (implies)g(\(RH\).)474 1564 y(Brou)m(w)m(er)44 b(logic)d(results)j (need)g(-)e(it)g(seems)i(-)e(some)h(explanations.)74 b(Firstly)-8 b(,)44 b(w)m(e)328 1684 y(prop)s(ose)26 b(the)g(follo)m(wing)c(explanation:)39 b(according)25 b(to)g(the)h(famous)f Fr(Go)s(del)j(theorem)328 1804 y Fq(on)34 b(the)h(non-solv)-5 b(alibit)m(y)31 b(of)j(axiomatic)e (theories)j([Go])e(w)m(e)j(cannot)e(pro)m(v)m(e)i(that)e(ev)m(en)328 1925 y(in)m(teger)g(arithmetic)f(is)h(not)g(con)m(tadictory)-8 b(.)50 b(Hence,)36 b(w)m(e)g(can)f(only)f(b)s(eliev)m(e)g(that)h(it)e (is)328 2045 y(so!)43 b(But)30 b(b)s(elief)e(is)h(only)g(b)s(elief)g(-) g(and)h(for)f(example)g(-)g(I)h(b)s(eliev)m(e)g(that)f(it)g(is)g(true)h (in)f(the)328 2166 y(case)f(of)e(arithmetic.)40 b(But)27 b(I)g(also)f(b)s(eliev)m(e)h(that)g(in)f(the)h(axiomatic)e(Zermelo-)g (F)-8 b(rank)m(el-)328 2286 y(Choice-axiom)25 b(\(ZF)m(C)i(in)g (short\))g(-)g(from)f(set)i(theory)g(-)f(\(a)g(con)m(tradiction)f(is)g (p)s(ossible\).)328 2406 y(The)44 b(parado)m(xes)g(of)e(Can)m(tor)h (and)g(Russel)h(in)e(early)g("naiv)m(e")h(and)g(non-)f(axiomatic)328 2527 y(Can)m(tor)h(set)g(theory)g(sho)m(ws)h(that)e(it)g(is)g(p)s (ossible.)72 b(The)43 b(consructions)g(from)f([M)s(\030)-46 b(a2])328 2647 y(and)34 b([M)s(\030)-46 b(a3])35 b(use)g(only)f (standard)h(maths)f(metho)s(ds)h(of)f(set)h(and)f(measure)h(theory)-8 b(.)50 b(So)328 2768 y(there)31 b(are)e(no)h(natural)f(reasons)i(to)e (reject)i(these)g(constuction)f(and)g(ma)m(yb)s(e)g(therefore)328 2888 y(is)37 b(no)m(w)h(time)d(to)i(ac)m(kno)m(wledge)i(that)e(Brou)m (w)m(er)h(logic)d(is)i(as)h(go)s(o)s(d)e(as)h(classical)f(logic)328 3008 y(\(as)f(w)m(as)h(done)g(-for)e(example)h(-in)f(informatics)f (\(cf.e.g.)52 b([De]\).)f(Ob)m(viously)35 b(w)m(e)h(ha)m(v)m(e)328 3129 y(to)f(pa)m(y)h(some)f(price)g(for)g(suc)m(h)i(a)e(w)m(eakness)j (of)d(classical)e(logic)h(:)49 b(\(RH\))35 b(is)f(b)s(oth)h(true)328 3249 y(and)i(false)g(!,)h(i.e.)57 b Fo(v)t Fq(\()p Fo(p)1159 3264 y Fk(RH)1279 3249 y Fn(^)36 b(\030)g Fo(p)1543 3264 y Fk(RH)1664 3249 y Fq(\))f(=)g(1,)j(where)h Fo(v)t Fq(\()p Fo(p)p Fq(\))d(denotes)j(the)e(classical)f(logic)328 3369 y(v)-5 b(alue)32 b(of)g(the)h(logical)d(statemen)m(t)j Fo(p)p Fq(.)474 3490 y(During)45 b(the)h(recen)m(t)h(edition)e(of)g (this)g(pap)s(er)h(M.)g(Bo)8 b(_)-35 b(zejk)m(o)47 b(informed)e(me)g (that)328 3610 y Fr(Matti)50 b(Pitk)-6 b(anen)43 b Fq([Pi])h(pro)m(v)m (ed)i(the)e Fr(Hilb)s(ert-P)m(oly)m(a)k(conjecture)d Fq(on)f Fo(\020)51 b Fq(using)328 3731 y(metho)s(ds)43 b(from)f(conformal)f(\014eld)i(theory)h(\(originated)d(in)i(quan)m(tum) g(ph)m(ysics\).)77 b(In)328 3851 y(his)37 b(pro)s(of)g(one)h(of)g(the)g (main)e(roles)h(is)h(pla)m(y)m(ed)g(b)m(y)h(the)f(function)f Fn(j)g Fo(z)k Fn(j)2996 3815 y Fe(2)3073 3851 y Fq(although)c(in)328 3971 y(this)28 b(case)h(as)f(eigen)m(v)-5 b(alues)29 b(of)e(the)i(\(semi-\))e(Hermitian)f(op)s(erator)h Fo(H)8 b Fq(,)29 b(whic)m(h)g(giv)m(es)f(the)328 4092 y(Hilb)s(ert-P)m(oly)m (a)34 b(conjecture.)56 b(An)37 b(idea)f(for)f(pro)m(ving)i(the)f (Riemann)f(h)m(yp)s(othesis)j(w)m(as)328 4212 y(to)24 b(giv)m(e)h(a)f Fr(sp)s(ectral)k(in)m(terpretetation)22 b Fq(of)i(the)h(zeros.)42 b(That)25 b(is,)h(if)d(the)i(zeros)h(can)f(b) s(e)328 4332 y(in)m(terpreted)30 b(as)f(the)g(eigen)m(v)-5 b(alues)29 b(of)f(1)p Fo(=)p Fq(2)15 b(+)g Fo(iT)f Fq(,)28 b(where)i Fo(T)43 b Fq(is)28 b(a)h Fr(Hermitian)d Fq(op)s(erator)328 4453 y(on)k(some)f(Hilb)s(ert)g(space,)i(then)g(since)f(the)g(zeros)h (of)e(a)h(Hermitian)d(op)s(erator)j(are)g(real,)328 4573 y(the)h(Riemann)e(h)m(yp)s(othesis)j(follo)m(ws.)41 b(This)31 b(idea)e(w)m(as)j(originally)27 b(put)j(forth)g(b)m(y)i(P)m(oly)m(a)328 4694 y(and)43 b(Hilb)s(ert)f(\(cf.e.g.)76 b([Ed]\),)47 b(and)c(serious)h(supp)s(ort)f(for)g(this)g(idea)g(w)m(as)h(found)g(in) 328 4814 y(the)h(resem)m(blence)h(b)s(et)m(w)m(een)g(the)f("explicit)e (form)m(ulae")g(of)h(prime)f(n)m(um)m(b)s(er)i(theory)-8 b(,)328 4934 y(whic)m(h)32 b(go)g(bac)m(k)h(to)e(Riemann)g(and)h(V)-8 b(on)32 b(Mangoldt,)f(but)h(whc)m(h)i(w)m(ere)f(formalized)d(as)1898 5214 y(10)p eop %%Page: 11 11 11 10 bop 328 631 a Fq(a)33 b(dualit)m(y)g(principle)f(b)m(y)j(W)-8 b(eil,)33 b(on)h(the)g(one)g(hand,)g(and)g(the)g(Selb)s(erg)g(trace)g (form)m(ula)328 751 y(on)e(the)h(other.)474 872 y(A)m(t)22 b(the)h(b)s(eginning)d(of)h(this)h(y)m(ear,)j(when)e(I)f(heard)g(a)g (short)g(measure)g(theoretic)g(pro)s(of)328 992 y(of)43 b(the)h(follo)m(wing)d(com)m(binatorial)g(conjecture)j(pro)m(v)m(ed)i (b)m(y)e(M.Mora)m(yne)i(in)c([Mo]:if)328 1112 y(the)47 b Fr(c)m(hromatic)52 b(n)m(um)m(b)s(er)47 b Fo(\037)p Fq(\()p Fm(R)1632 1076 y Fe(2)1672 1112 y Fq(\))f(of)h(the)g(complex)f (plane)g(is)h(equal)f(7,then)h(the)328 1233 y(underlying)32 b(graph)h(m)m(ust)g(ha)m(v)m(e)i(6800)d(ap)s(exes,)i(then)g(I)f (realized)f(that)h(if)f(there)i(exists)328 1353 y(a)c(short)g(pro)s(of) g(of)g(\(RH\),)g(then)h(it)e(is)h(p)s(ossible)f(on)h(the)h(ground)f(of) g(the)h(classical)e(logic.)328 1474 y(In)23 b(this)f(case)h(w)m(e)g(do) g(not)f(need)h(the)g(complicated)e(mac)m(hinery)h(of)g(Ho)s(dge)g (measures)i(and)328 1594 y(decomp)s(ositions,)k(based)h(on)f(set)h (theory)-8 b(,)30 b(whic)m(h)e(can)h(leads)f(to)g(a)g(con)m(tradiction) f(\(and)328 1714 y(in)34 b(addition)f(w)m(e)j(m)m(ust)f(to)g(w)m(ork)g (on)g(the)g(ground)g(of)g(Brou)m(w)m(er)h(logic\).)48 b(In)35 b(F)-8 b(ebruary)g(,)328 1835 y(the)30 b(p)s(ossibilit)m(y)e (of)h(suc)m(h)j(a)d(pro)s(of)g(w)m(as)i(sho)m(w)m(ed)h(b)m(y)f(Matti)e (Pitk)-5 b(anen)30 b(in)f([Pi].)42 b(So,)30 b(the)328 1955 y(pap)s(er)j([Mo])f(w)m(as)i(the)f(direct)f(motiv)-5 b(ation)30 b(of)i(this)g(article.)474 2076 y(According)41 b(to)g Fr(P)m(oincare)f Fq([P)m(o])h(the)h(unique)f(thing,)h(whic)m(h)g (w)m(e)g(m)m(ust)f(demand)328 2196 y(from)k(an)i(ob)5 b(ject)48 b(whic)m(h)f(exists)h(in)e(mathematics)f(is)h Fr(non-con)m(tradictivit)m(y)e Fq(\(al-)328 2316 y(though)39 b(in)e(the)i(ligh)m(t)e(of)h(resen)m(t)i(results)f(it)f(is)g(also)g (problematic\).)59 b(Moreo)m(v)m(er)40 b(that)328 2437 y(requiremen)m(t)35 b(could)g(not)g(b)s(e)g(k)m(ept,)i(as)e(in)f(the)h (ligh)m(t)e(of)i(Go)s(del's)f(results,)i(it)e(w)m(as)i(not)328 2557 y(p)s(ossible)i(to)h(k)m(eep)h(the)g(requiremen)m(t)f(of)f(the)i (completness)f(of)g(the)g(ma)5 b(jorit)m(y)38 b(of)g(ax-)328 2677 y(iomatic)30 b(systems.)474 2798 y(In)i(the)f(approac)m(h)h(of)e (P)m(oincare)h(men)m(tioned)g(ab)s(o)m(v)m(e,)h(the)g(Can)m(tor)f(set)h (theory)g(w)m(as)328 2918 y("bad",)43 b(b)s(ecause)g(it)e(lead)g(to)g (the)h(w)m(ell-kno)m(wn)g(con)m(tradictions)e(:)62 b(the)42 b(Can)m(tor)g(and)328 3039 y(Russell)f(parado)m(xes,)k(")c(but)h(con)m (tradictions)f(whic)m(h)h(can)g(exp)s(ose)h(themselv)m(es)g(-)e(w)m(e) 328 3159 y(cannot)h(forecast"[P)m(o,)i(Bo)s(okI)s(I.)d(Section)g(I)s(I) s(I\(Maths)i(and)e(Logic.)69 b(In)m(tro)s(duction\)].)328 3279 y(Ob)m(viously)-8 b(,)46 b(if)c(the)h(early)g(Can)m(tor)h(set)g (theory)g(is)e(not)h(bad,)k(then)c(w)m(e)i(m)m(ust)e(reject)328 3400 y(TnD)30 b(-)f(what)h(Brou)m(w)m(er)h(has)f(done)g(\(for)f(man)m (y)h(Brou)m(w)m(er)g(has)h(p)s(oured)e(out)h("the)g(bab)m(y)328 3520 y(with)i(the)h(w)m(ater"!\).)474 3641 y(Ob)m(viously)d(ZF)m (C-axioms)e(remo)m(v)m(e)j(con)m(tradictions)e(men)m(tioned)g(ab)s(o)m (v)m(e)i(\()f(of)f(Can-)328 3761 y(tor)41 b(and)h(Russel)g(t)m(yp)s (e\))g(but)g(w)m(e)h(do)e(not)h(kno)m(w)g(if)f(some)g(new)i(parado)m (xes)g("do)e(not)328 3881 y(exp)s(ose)34 b(themselv)m(es")f([P)m(o].) 474 4002 y(No)m(w)h(\()f(at)h(the)f(b)s(eginning)f(of)h(3rd)g (millennium\))d(w)m(e)k(kno)m(w)g(,)g(that)f(it)g(really)f(is)h(so)328 4122 y(-)f(e.g.)44 b(the)33 b(fundamen)m(tal)e(examples)i(men)m(tioned) f(here)h(from)f(constructiv)m(e)i(maths.)474 4242 y(I)41 b(think)g(that)g(one)g(of)f(the)h(reasons)h(that)f(\(RH\))f(w)m(as)i (an)f(op)s(en)g(problem)e(for)i(so)328 4363 y(long)33 b(-)g(is)g(the)h(problems)g(considered)g(ab)s(o)m(v)m(e,)h(together)f (with)g(the)g(famous)f("crisis)g(of)328 4483 y(fundamen)m(ts)40 b(of)f(maths")f(since)i(the)g(b)s(eginning)e(of)g(the)i(XX-cen)m(taury) -8 b(.)65 b(This)39 b(crisis)328 4604 y(has)33 b(still)d(not)j (expired.)474 4724 y(A)m(t)40 b(the)g(end)h(of)e(this)g(long)f(in)m (tro)s(duction)h(w)m(e)h(will)e(try)i(also)e(to)i(explain)e(wh)m(y)k (in)328 4844 y(the)36 b(title)e(of)h(this)h(pap)s(er)g(the)g(w)m(ord)g ("algebraic")e(\014gures.)54 b(That)36 b(is)f(in)g(con)m(tadiction)328 4965 y(to)j(the)h(completely)e(analytic)g(pro)s(of)h(of)g(\(RH\))g(of)g (Matti)f(Pitk)-5 b(anen)39 b([Pi],)g(our)g(pro)s(of)1898 5214 y(11)p eop %%Page: 12 12 12 11 bop 328 631 a Fq(uses)49 b(purely)g Fr(algebraic)54 b(ob)6 b(jects)49 b Fq(suc)m(h)g(as)f(:)75 b Fo(I)8 b Fq(\()p Fo(s)p Fq(\))p Fo(;)48 b(s)p Fq(\()p Fo(s)32 b Fn(\000)h Fq(1\))p Fo(;)17 b Fn(j)53 b Fo(:)i Fn(j)3068 595 y Fe(2)3107 631 y Fo(;)17 b(Gal)r Fq(\()p Fm(C)23 b Fo(=)p Fm(R)9 b Fq(\))328 751 y(and)47 b Fr(Cramer)52 b(systems)p Fq(.)86 b(Moreo)m(v)m(er)48 b(it)d(also)h(refers)h(to)f(-)g (in)g(some)g(sense)i(-)e(the)328 872 y(purely)22 b(algebraic)f (\(cohomological\))d(pro)s(of)k(of)f(the)i Fr(W)-9 b(eil)23 b(conjectures)g Fq(\()f(the)h(Dw)m(ork-)328 992 y(Grothendiec)m(k)33 b(and)g(Deligne's)e(theorems\).)328 1322 y Fp(2)161 b(Three)35 b(imp)t(ortan)l(t)g(transforms)g(:)62 b(Mellin,)41 b(F)-13 b(ourier)570 1504 y(and)39 b Fc(\022)k Fp(asso)t(ciated)c(with)g(the)f (Riemann)h(h)l(yp)t(oth-)570 1687 y(esis)328 1906 y Fq(F)-8 b(or)28 b(a)g(large)g(class)h(of)f(\000)p Fr(-admissible)f Fq(functions)i Fo(f)38 b Fq(:)28 b Fm(R)2399 1921 y Fe(+)2485 1906 y Fn(\000)-16 b(!)28 b Fm(C)52 b Fq(the)29 b Fr(Mellin)i(trans-) 328 2026 y(form)h Fq(M\(f)7 b(\))32 b(\(or)h Fr(Gamma)k Fq(\000\()p Fo(f)11 b Fq(\))37 b Fr(asso)s(ciated)h(with)e Fo(f)48 b Fq(\))32 b(is)g(w)m(ell-de\014ned)h(as)655 2268 y(\000\()p Fo(f)11 b Fq(\)\()p Fo(s)p Fq(\))55 b(:=)1158 2151 y Ff(Z)1241 2177 y Fe(+)p Fl(1)1205 2340 y Fe(0)1388 2268 y Fo(x)1443 2227 y Fk(s)p Fl(\000)p Fe(1)1570 2268 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))p Fo(dx)56 b Fq(=:)f Fo(M)10 b Fq(\()p Fo(f)h Fq(\)\()p Fo(s)p Fq(\))55 b(;)17 b Fo(R)q(e)p Fq(\()p Fo(s)p Fq(\))28 b Fo(>)f Fq(0)p Fo(:)328 b Fq(\(2.25\))328 2500 y(\(The)45 b Fr(Sc)m(h)m(w)m(artz)50 b(space)45 b Fn(S)7 b Fq(\()p Fm(R)j Fq(\))44 b(of)f(rapidly)g (decreasing)i(functions)f(constitutes)h(a)328 2621 y(small)30 b(class)j(of)f(\000-admissible)e(functions\).)474 2741 y(Recall)h(that)i Fo(f)38 b Fq(:)28 b Fm(R)36 b Fn(\000)-16 b(!)27 b Fm(C)56 b Fq(b)s(elongs)33 b(to)f Fn(S)7 b Fq(\()p Fm(R)i Fq(\),)33 b(i\013)e(for)h(eac)m(h)i(p)s(ositiv)m(e)e(in)m(teger) g Fo(k)s(;)17 b(l)1160 2939 y(p)1209 2954 y Fk(k)r(;l)1293 2939 y Fq(\()p Fo(f)11 b Fq(\))55 b(:=)g Fo(sup)1792 2968 y Fk(x)p Fl(2)p Fm(R)1976 2939 y Fn(j)27 b Fo(x)2086 2898 y Fk(k)2129 2939 y Fo(f)2188 2898 y Fe(\()p Fk(l)q Fe(\))2269 2939 y Fq(\()p Fo(x)p Fq(\))h Fn(j)f Fo(<)55 b Fq(+)p Fn(1)p Fo(:)328 3137 y Fq(The)40 b Fr(F)-9 b(rec)m(het)44 b(space)h(top)s(ology)38 b Fq(on)h Fn(S)7 b Fq(\()p Fm(R)i Fq(\))39 b(is)f(giv)m(en)h(b)m(y)h(a)e(family)f(of)h(seminorms)328 3258 y Fn(f)p Fo(p)427 3273 y Fk(k)r(;l)539 3258 y Fq(:)27 b Fo(k)s(;)17 b(l)30 b Fn(2)e Fm(N)13 b Fn(g)27 b Fq(\(it)g(is)g(a)g 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4626 y(0)660 4555 y Fn(j)28 b Fo(f)11 b Fq(\()p Fo(x)p Fq(\))p Fo(x)961 4514 y Fe(\()p Fk(s)p Fl(\000)p Fe(1\))1171 4555 y Fn(j)27 b Fo(dx)h Fn(\024)g Fo(p)1514 4570 y Fk(u)p Fl(\000)p Fk(\017;)p Fe(0)1697 4555 y Fq(\(=)g Fo(sup)1990 4583 y Fk(x)p Fl(2)p Fm(R)2173 4555 y Fn(j)g Fo(x)2284 4514 y Fk(u)p Fl(\000)p Fk(\017)2413 4555 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))27 b Fn(j)p Fq(\))2713 4438 y Ff(Z)2796 4464 y Fe(1)2759 4626 y(0)2898 4487 y Fo(dx)p 2862 4532 179 4 v 2862 4623 a(x)2917 4594 y Fe(1)p Fl(\000)p Fk(\017)3077 4555 y Fo(<)h Fq(+)p Fn(1)p Fo(;)328 4782 y Fq(and)904 4809 y Ff(Z)987 4836 y Fe(+)p Fl(1)951 4998 y Fe(1)1145 4926 y Fn(j)f Fo(f)11 b Fq(\()p Fo(x)p Fq(\))p Fo(x)1445 4885 y Fk(s)p Fl(\000)p Fe(1)1600 4926 y Fn(j)27 b Fo(dx)h(<)55 b(p)1969 4941 y Fk(u)p Fe(+)p Fk(\017;)p Fe(0)2153 4926 y Fq(\()p Fo(f)11 b Fq(\))2305 4809 y Ff(Z)2387 4836 y Fe(+)p Fl(1)2350 4998 y Fe(1)2579 4859 y Fo(dx)p 2543 4903 V 2543 4995 a(x)2598 4966 y Fe(1+)p Fk(\017)2759 4926 y Fo(<)28 b Fn(1)p Fo(;)1898 5214 y Fq(12)p eop %%Page: 13 13 13 12 bop 328 631 a Fq(if)31 b Fo(R)q(e)p Fq(\()p Fo(s)p Fq(\))d Fo(>)g Fq(0.)474 751 y(Also)j(for)f(another)h(large)f(class)h (of)f Fo(\022)s Fr(-admissible)g Fq(functions)h Fo(f)39 b Fq(:)27 b([1)p Fo(;)17 b Fq(+)p Fn(1)p Fq(\))27 b Fn(\000)-16 b(!)27 b Fm(C)328 872 y Fr(theta)37 b(function)32 b Fo(\022)s Fq(\()p Fo(f)11 b Fq(\))33 b Fr(asso)s(ciated)38 b(with)31 b Fo(f)43 b Fq(is)32 b(de\014ned)i(as)f(the)g(series)890 1139 y Fo(\022)s Fq(\()p Fo(f)11 b Fq(\)\()p Fo(x)p Fq(\))55 b(:=)1421 1031 y Fe(+)p Fl(1)1424 1056 y Ff(X)1417 1238 y Fk(n)p Fe(=1)1567 1139 y Fo(f)11 b Fq(\()p Fo(nx)p Fq(\))56 b(=)2002 1022 y Ff(Z)2048 1211 y Fm(N)2114 1169 y Fd(\003)2170 1139 y Fo(f)11 b Fq(\()p Fo(nx)p Fq(\))p Fo(dc)p Fq(\()p Fo(x)p Fq(\))56 b(;)17 b Fo(x)28 b(>)f Fq(0)p Fo(;)313 b Fq(\(2.26\))328 1401 y(where)34 b Fo(dc)e Fq(denotes)i(the)f Fr(calculating)i(measure)e Fq(on)g Fm(N)2455 1365 y Fl(\003)2495 1401 y Fq(.)474 1522 y(By)38 b Fn(F)10 b Fo(f)46 b Fq(w)m(e)38 b(denote)g(the)f Fr(F)-9 b(ourier)41 b(transform)c Fq(of)f Fo(f)48 b Fq(\(for)36 b Fn(F)10 b Fq(-admissible)34 b(func-)328 1642 y(tions\):)956 1793 y Fn(F)10 b Fq(\()p Fo(f)h Fq(\)\()p Fo(x)p Fq(\))54 b(:=)1517 1675 y Ff(Z)1563 1864 y Fm(R)1649 1793 y Fo(e)1694 1751 y Fe(2)p Fk(\031)r(ixy)1878 1793 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)57 b Fq(=:)2400 1766 y(^)2379 1793 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))55 b(;)17 b Fo(x)28 b Fn(2)g Fm(R)9 b Fo(:)379 b Fq(\(2.27\))474 1992 y(The)41 b(Sc)m(h)m(w)m(artz)i(space)e Fn(S)7 b Fq(\()p Fm(R)i Fq(\))41 b(is)e Fr(admissible)g Fq(for)h(the)g(all)f(in)m(tegral)f (transforms)328 2112 y(considered)33 b(ab)s(o)m(v)m(e:)45 b Fo(M)5 b(;)17 b(\022)s(;)g Fn(F)10 b Fq(.)474 2353 y(In)26 b(this)f(pap)s(er)g(more)g(con)m(v)m(enien)m(t)i(will)c(b)s(e)i (w)m(ork)h(with)f(the)h Fr(plus-F)-9 b(ourier)28 b(trans-)328 2473 y(form)37 b Fn(F)667 2488 y Fe(+)758 2473 y Fq(de\014ned)d(as)383 2713 y Fn(F)455 2728 y Fe(+)514 2713 y Fq(\()p Fo(f)11 b Fq(\)\()p Fo(x)p Fq(\))55 b(:=)993 2595 y Ff(Z)1077 2622 y Fe(+)p Fl(1)1040 2784 y Fe(0)1223 2713 y Fo(e)1268 2671 y Fe(2)p Fk(\031)r(ixy)1451 2713 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)57 b Fq(=:)1946 2686 y(^)1925 2713 y Fo(f)1973 2728 y Fe(+)2032 2713 y Fq(\()p Fo(x)p Fq(\))28 b(;)44 b Fo(x)28 b Fn(2)h Fm(R)2505 2728 y Fe(+)2564 2713 y Fo(:)726 b Fq(\(2.28\))328 3042 y Fp(3)161 b(Quasi-\014xed)54 b(p)t(oin)l(ts)f(of)g Fa(F)69 b Fp(and)53 b(\(RH\))328 3261 y Fq(This)30 b(section)h(underline)e(the)i(role)e(and)h(imp)s (ortance)f(of)h(the)g(nation)f(of)h(a)g(quasi-\014xed)328 3382 y(p)s(oin)m(t)i(of)g Fn(F)42 b Fq(and)32 b(the)h(Riemann)f(h)m(yp) s(othesis)h(problem.)474 3502 y(A)46 b(complex)f(function)h Fo(!)j Fq(on)c Fm(R)55 b Fq(is)45 b(called)g(a)g Fr(\014xed)54 b(p)s(oin)m(t)d(of)46 b Fn(F)10 b Fq(,)49 b(if)c(it)g(is)g(an)328 3622 y Fr(eigen)m(v)m(ector)31 b Fq(of)i Fn(F)42 b Fq(with)32 b(an)g Fr(eigen)m(v)-6 b(alue)32 b Fq(equal)h(to)f(1,)g(i.e.)1613 3821 y Fn(F)10 b Fq(\()p 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Fk(\031)r(\033)r(x)2411 4795 y Fj(2)2451 4855 y Fq(\))p 1505 4899 984 4 v 1800 4990 a(1)g(+)g Fo(\033)2028 4962 y Fl(\000)p Fe(1)p Fk(=)p Fe(2)2526 4922 y Fq(;)45 b Fo(x)28 b Fn(2)g Fm(R)8 b Fo(:)450 b Fq(\(3.30\))1898 5214 y(13)p eop %%Page: 14 14 14 13 bop 328 631 a Fq(W)-8 b(e)29 b(see)h(that)e Fo(G)930 646 y Fk(\033)977 631 y Fq(\(0\))f(=)h(1)p Fo(;)17 b(G)1403 646 y Fk(\033)1477 631 y Fn(2)28 b(S)7 b Fq(\()p Fm(R)i Fq(\))29 b(and)f Fn(F)10 b Fq(.)42 b(But,)30 b(subsequen)m(tly)-8 b(,)32 b Fo(G)3034 646 y Fk(\033)3109 631 y Fq(is)c(a)h(sp)s(ecial)328 751 y(case)d(of)e(the)i(more)e(general)g(construction,)j(whic)m(h)f(is) e(based)i(on)f(the)g(WELL-KNO)m(WN)328 872 y(fact)36 b(that)g(F)-8 b(ourier)35 b(transform)g(is)h(an)g Fr(idemp)s(oten)m(t)f Fq(\(a)h(part)g(of)f Fr(Planc)m(herel)40 b(the-)328 992 y(orem)p Fq(\))32 b(on)h(the)g(class)g(of)f Fn(F)42 b Fq(-admissible)31 b(symmetric)h(functions.)44 b(More)33 b(exactly)-8 b(,if)32 b Fo(f)328 1112 y Fq(is)j(a)f Fn(F)10 b Fq(-admissible)33 b(and)i Fr(symmetric)e Fq(\(i.e.)50 b Fo(f)11 b Fq(\()p Fn(\000)p Fo(x)p Fq(\))33 b(=)f Fo(f)11 b Fq(\()p Fo(x)p Fq(\)\))35 b(function)g(on)g Fm(R)8 b Fq(,)36 b(then)328 1233 y(according)c(to)g(the)h Fr(Planc)m(herel)j (theory)p Fq(9cf.e.g.)44 b([Ha,)32 b(Th.3.13])1689 1453 y Fo(!)59 b Fq(:=)c Fo(f)61 b Fq(+)2223 1427 y(^)2202 1453 y Fo(f)10 b(;)1030 b Fq(\(3.31\))328 1673 y(is)22 b(a)h Fr(\014xed)k(p)s(oin)m(t)c Fq(of)f Fn(F)10 b Fq(.)40 b(The)24 b(de\014nition)e(\(?\))40 b(is)23 b(a)f(sp)s(ecial)h(case)g (of)g(the)g(fundamen)m(thal)328 1793 y(nation)38 b(whic)m(h)h(w)m(e)g (are)g(going)e(to)i(explore)g(in)f(this)g(pap)s(er.)62 b(Jest)39 b(to)g(na)5 b(jistotniejsze)328 1914 y(uogolnienie,)27 b("gw)m(ozdz)i(program)m(u",)f(esencja)h(\(ekstrak)m(cja\)z)h(\014xed)g (punktu)f(W)-8 b(e)28 b(pro-)328 2034 y(p)s(ose)33 b(the)g(follo)m (wing)d(de\014nition)328 2237 y Fr(De\014nition)36 b(3.1)49 b Fb(\(Of)34 b(a)h(quasi-\014xe)-5 b(d)34 b(p)-5 b(oint)34 b(of)h Fn(F)10 b Fb(\).)328 2358 y(L)-5 b(et)41 b Fo(f)49 b Fq(:)38 b Fm(R)47 b Fn(\000)-16 b(!)38 b Fm(C)65 b Fb(b)-5 b(e)40 b Fr(symmetric)e Fb(and)i Fn(F)10 b Fb(-admissible.)60 b(L)-5 b(et)41 b Fo(\025)d Fn(2)h Fm(C)64 b Fb(b)-5 b(e)40 b(arbitr)-5 b(ary)328 2478 y(p)g(ar)g(ameter.)51 b(We)37 b(say)g(that)h(a)e(function)h Fo(F)1923 2493 y Fk(\025)2001 2478 y Fq(=)31 b Fo(Q)2185 2493 y Fk(\025)2231 2478 y Fq(\()p Fo(f)11 b Fq(\))31 b(:)h Fm(R)41 b Fn(\000)-17 b(!)32 b Fm(C)61 b Fb(is)36 b(a)h Fr(quasi-\014xed)328 2599 y(p)s(oin)m(t)g(of)g(F)f Fb(\(asso)-5 b(ciate)g(d)33 b(with)i Fo(f)46 b Fb(and)34 b(the)h Fr(sp)s(ectral)f Fb(p)-5 b(ar)g(ameter)34 b Fo(\025)p Fb(\))h(i\013)998 2819 y Fq(\()p Fo(QF)14 b Fq(\))55 b Fo(F)1346 2834 y Fk(\025)1392 2819 y Fq(\()p Fo(x)p Fq(\))g(=)g Fo(Q)1786 2834 y Fk(\025)1832 2819 y Fq(\()p Fo(f)11 b Fq(\)\()p Fo(x)p Fq(\))55 b(:=)h Fo(f)11 b Fq(\()p Fo(x)p Fq(\))22 b(+)g Fo(\025)2700 2792 y Fq(^)2679 2819 y Fo(f)10 b Fq(\()p Fo(x)p Fq(\))p Fo(;)422 b Fq(\(3.32\))328 3039 y Fo(x)28 b Fn(2)g Fm(R)9 b Fb(.)328 3267 y Fr(Remark)37 b(1)49 b Fb(L)-5 b(et)28 b Fo(!)k Fb(b)-5 b(e)28 b(a)h Fr(\014xed)h(p)s(oin)m(t)d Fb(of)i Fn(F)10 b Fb(.)42 b(Then)27 b(obviously)h Fo(!)k Fb(is)c(a)g Fr(quasi-\014xed)328 3387 y(p)s(oin)m(t)34 b Fb(of)h Fn(F)44 b Fb(:)1268 3552 y Fo(!)59 b Fq(=)c Fo(Q)1596 3567 y Fe(1)1636 3552 y Fq(\()35 b Fo(omeg)t(a=)p Fq(2\))54 b(=)2320 3484 y Fo(!)p 2320 3528 65 4 v 2328 3620 a Fq(2)2416 3552 y(+)2532 3484 y(^)-57 b Fo(!)p 2524 3528 V 2532 3620 a Fq(2)2598 3552 y Fo(:)328 3751 y Fb(Thus)32 b(we)h(se)-5 b(e)32 b(that)h(the)g(notion)f(of)h(a)g(quasi-\014xe)-5 b(d)31 b(p)-5 b(oint)33 b(is)f(a)h(dir)-5 b(e)g(ct)32 b(extension)g(of)h(the) 328 3872 y(notion)h(of)h(a)g(\014xe)-5 b(d)34 b(p)-5 b(oint.)328 4100 y Fq(Quasi-\014xed)33 b(p)s(oin)m(ts)f(ha)m(v)m(e)i (got)e(a)g(lot)f(of)h(nice)h(algebraic)e(prop)s(erties:)328 4220 y(\(P1\))i Fo(Q)629 4235 y Fe(1)669 4220 y Fq(\()p Fo(f)11 b Fq(\))33 b(is)g(alw)m(a)m(ys)h(a)f(\014xed)i(p)s(oin)m(t)e 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Fq(=)g Fo(F)3467 4759 y Fk(\025)3539 4744 y Fq(.)328 4864 y(By)i Fo(Q)p Fq(\012)h(w)m(e)f(denote)h(the)f(set)h(of)e(all)f Fr(quasi-\014xed)34 b(p)s(oin)m(ts)28 b Fq(of)h Fn(F)38 b Fq(and)29 b(b)m(y)g(\012)h(the)f(set)g(of)1898 5214 y(14)p eop %%Page: 15 15 15 14 bop 328 631 a Fq(all)23 b(\014xeed)k(p)s(oin)m(ts)e(whic)m(h)h (satis\014es)g(the)g(assumptions)f(of)g(the)h Fr(P)m(oisson)i (Summation)328 751 y(F)-9 b(orm)m(ula)32 b Fq(\(PSF)g(in)g(short)h (\(cf.e.g.[Ch,I)s(I,Th.1],)i([Ma,Th.XVI)s(I)s(I.26],[Na],[F)-8 b(e]\)\))1269 960 y(\()p Fo(P)14 b(S)6 b(F)14 b Fq(\))1706 877 y Ff(X)1691 1078 y Fk(n)p Fl(2)p Fm(Z)1858 960 y Fo(f)d Fq(\()p Fo(n)p Fq(\))55 b(=)2253 877 y Ff(X)2238 1078 y Fk(n)p Fl(2)p Fm(Z)2426 933 y Fq(^)2405 960 y Fo(f)10 b Fq(\()p Fo(n)p Fq(\))p Fo(:)328 1265 y Fr(Remark)37 b(2)49 b Fb(L)-5 b(et's)26 b(c)-5 b(onsider)25 b(the)i(Banach)e(sp)-5 b(ac)g(e)25 b Fo(l)2237 1229 y Fe(1)2277 1265 y Fq(\()p Fm(R)8 b Fq(\))27 b Fb(of)f(al)5 b(l)25 b(absolutely)i(c)-5 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Fk(\025)1540 2891 y Fq(\()p Fo(xy)t Fq(\);)17 b Fo(x)27 b(>)h Fq(0)p Fo(;)17 b(y)30 b Fn(2)e Fm(R)9 b Fq(,)30 b(in)e(the)i(PSF,)f(and)h(observ)m(e)h(that) 820 3123 y(^)784 3148 y Fo(F)861 3114 y Fk(x)847 3174 y(\025)905 3148 y Fq(\()p Fo(y)t Fq(\))54 b(=)1218 3031 y Ff(Z)1264 3220 y Fm(R)1351 3148 y Fo(e)1396 3107 y Fe(2)p Fk(\031)r(iy)r(z)1575 3148 y Fo(F)1638 3163 y Fk(\025)1684 3148 y Fq(\()p Fo(xz)t Fq(\))p Fo(dz)60 b Fq(=)c Fn(f)p Fo(xz)32 b Fq(=)c Fo(w)s(;)17 b(dz)31 b Fq(=)2796 3081 y Fo(dw)p 2796 3125 124 4 v 2829 3217 a(x)2929 3148 y Fn(g)55 b Fq(=)1179 3448 y(=)1296 3380 y(1)p 1292 3425 56 4 v 1292 3516 a Fo(x)1374 3331 y Ff(Z)1420 3519 y Fm(R)1507 3448 y Fo(e)1552 3407 y Fe(2)p Fk(\031)r(i)1664 3375 y Fg(y)r(w)p 1664 3392 80 4 v 1686 3433 a(x)1758 3448 y Fo(F)1821 3463 y Fk(\025)1866 3448 y Fq(\()p Fo(w)s Fq(\))p Fo(dw)i Fq(=)2354 3423 y(^)2324 3448 y Fo(F)2387 3463 y Fk(\025)2432 3448 y Fq(\()2482 3380 y Fo(y)p 2480 3425 56 4 v 2480 3516 a(x)2546 3448 y Fq(\))p Fo(=x;)474 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b(:=)1862 4784 y Fe(+)p Fl(1)1865 4808 y Ff(X)1859 4990 y Fk(n)p Fe(=1)2008 4891 y Fo(F)2071 4906 y Fk(\025)2117 4891 y Fq(\()p Fo(nx)p Fq(\);)72 b Fo(x)28 b(>)g Fq(0)p Fo(;)649 b Fq(\(3.35\))1898 5214 y(15)p eop %%Page: 16 16 16 15 bop 328 631 a Fq(and)1388 796 y Fo(\022)s Fq(\()1504 770 y(^)1474 796 y Fo(F)1537 811 y Fk(\025)1582 796 y Fq(\)\()p Fo(x)p Fq(\))1800 688 y Fe(+)p Fl(1)1802 713 y Ff(X)1796 894 y Fk(n)p Fe(=1)1975 770 y Fq(^)1945 796 y Fo(F)2008 811 y Fk(\025)2054 796 y Fq(\()p Fo(nx)p Fq(\))56 b Fo(x)28 b(>)f Fq(0)p Fo(;)756 b Fq(\(3.36\))328 1039 y(then)33 b(w)m(e)h(obtain)d(the)i(\()p Fo(\022)s(:f)5 b(:e)p Fq(\).)474 1159 y(In)31 b(the)g(case)g(of)f(a)g(\014xed)i(p)s (oin)m(t)d Fo(!)i Fq(=)d Fo(Q)1911 1174 y Fe(1)1951 1159 y Fq(\()p Fo(f)11 b Fq(\))30 b(the)h(result)f(of)g(the)h(Lemma)e(2)h (reduces)328 1280 y(to)i(the)h(famous)f Fr(classical)k Fo(\022)s Fr(-equation)d Fq(of)f(the)h(form)1234 1539 y Fo(\022)s Fq(\()p Fo(!)t Fq(\)\()p Fo(x)p Fq(\))55 b(=)1778 1471 y(1)p 1750 1515 105 4 v 1750 1607 a(2)p Fo(x)1864 1539 y Fq(\(2)p Fo(\022)s Fq(\()p Fo(!)t Fq(\)\()2191 1471 y(1)p 2188 1515 56 4 v 2188 1607 a Fo(x)2253 1539 y Fq(\))22 b(+)g(1\))g Fn(\000)2629 1471 y Fq(1)p 2629 1515 49 4 v 2629 1607 a(2)2688 1539 y Fo(:)328 1790 y Fq(Let's)28 b(write)g(the)g Fr(Chandrasekharan's)35 b(b)s(o)s(ok)d ([CH])g(form)27 b Fq(of)g(\(R.f.e.\))42 b(using)27 b(our)328 1910 y(notations)32 b(from)f(the)i(previous)g(section)g(:)328 2180 y(\()p Fo(C)7 b(H)r(:R)q(:f)e(:e)p Fq(\))56 b(\(\000\()p Fo(G)p Fq(\))p Fo(\020)8 b Fq(\)\()p Fo(s)p Fq(\))54 b(=)1625 2112 y Fo(G)p Fq(\(0\))p 1532 2156 388 4 v 1532 2248 a(2)p Fo(s)p Fq(\()p Fo(s)22 b Fn(\000)h Fq(1\))1929 2180 y(+)2005 2062 y Ff(Z)2088 2089 y Fe(+)p Fl(1)2051 2251 y Fe(1)2218 2180 y Fq(\()p Fo(x)2311 2138 y Fk(s)p Fl(\000)p Fe(1)2438 2180 y Fq(+)p Fo(x)2569 2138 y Fl(\000)p Fk(s)2661 2180 y Fq(\))p Fo(\022)s Fq(\()p Fo(G)p Fq(\)\()p Fo(x)p Fq(\))p Fo(dx)p Fq(;)72 b Fo(s)28 b Fn(2)g Fm(C)c Fn(\000f)p Fq(0)p Fo(;)17 b Fq(1)p Fn(g)p Fo(:)3317 2356 y Fq(\(3.37\))328 2476 y(It)38 b(w)m(ell-kno)m(wn)g(that)g (\(CH.R.f.e.\))60 b(is)38 b(the)g(immediate)e(consequence)41 b(of)c(the)i(\(PSF\))328 2597 y(and)33 b(the)g(iden)m(tit)m(y)f(:)1139 2571 y(^)1116 2597 y Fo(G)c Fq(=)g Fo(G)p Fq(.)474 2717 y(Replacing)33 b Fo(G)g Fq(b)m(y)i(an)e(arbitrary)g(\014xed)i(p)s(oin)m (t)41 b(^)-57 b Fo(!)33 b Fq(=)c Fo(!)37 b Fq(of)c Fn(F)44 b Fq(for)33 b(whic)m(h)h(holds)f(the)328 2837 y(ab)s(o)m(v)m(e)f (classical)e Fo(\022)s Fq(-equation)h(w)m(e)i(obtain)d(the)i(extension) g(of)f(\(CH.R.f.e\))i(to)e(this)g(case)328 2958 y(\(cf.)42 b(the)29 b(pro)s(of)f(of)g(Theorem)g(in)g([CH]\).)h(But)g(ev)m(en)h (the)f(in\015ation)d(\(spliting\))h(of)h(\(3.37\))328 3078 y(to)c(the)g(whole)g(class)g(\012)g(\014xed)h(p)s(oin)m(ts)f(of)g Fn(F)33 b Fq(is)23 b(not)h(su\016ecien)m(t!)42 b(The)25 b(reason)g(is)e(that)h(-)g(so)328 3198 y(called)32 b(-)g (\(RH\)-\014xed)i(p)s(oin)m(ts)e(of)h Fn(F)42 b Fq(do)33 b(not)g(exist!)45 b(More)33 b(exactly)-8 b(,)33 b(the)h(fundamen)m(tal) 328 3319 y(for)f(the)h(pro)s(of)f(of)g(\(RH\))g(-)h(considered)g(b)s (elo)m(w)g(-)f(the)h Fr(F)-9 b(o)m(x)39 b(in)m(tegral)e(equation)c Fq(\(i.e.)328 3439 y(some)f(sp)s(ecial)f Fr(F)-9 b(redholm)32 b Fq(equation)g(of)g(second)h(order\))f(has)h(only)f(solutions)f(on)h (the)328 3560 y(lev)m(el)42 b(of)h(quasi-\014xed)g(p)s(oin)m(ts.)74 b(These)45 b(solutions)c Fo(F)2315 3575 y Fk(\025)2404 3560 y Fq(w)m(e)j(call)d Fr(\(RH\)-quasi-\014xed)328 3680 y(p)s(oin)m(ts)35 b(of)h Fn(F)k Fq(and)31 b(exactly)h Fo(F)1487 3695 y Fe(1)1557 3680 y Fq(\(b)s(eing)e(the)i(\014xed)g(p)s (oin)m(t\))e(is)h(the)g Fr(singular)f Fq(solution)328 3800 y(of)g(the)i(men)m(tioned)f(ab)s(o)m(v)m(e)g(F)-8 b(o)m(x)31 b(equation.)43 b(Th)m(us,)33 b(fortunately)e(for)f(us)i(-)e (in)g(the)i(large)328 3921 y(class)f(of)g(quasi-\014xed)h(p)s(oin)m(ts) e(of)h Fn(F)41 b Fq(denoted)32 b(here)g(b)m(y)g Fo(Q)p Fq(\012,)g(there)g(exist)f(\(RH\)-quasi-)328 4041 y(\014xed)j(p)s(oin)m (ts)e Fo(F)920 4056 y Fk(\025)993 4041 y Fq(=)c Fo(f)k Fq(+)22 b Fo(\025)1354 4015 y Fq(^)1332 4041 y Fo(f)11 b Fq(.)474 4162 y(The)33 b(\()p Fo(\022)s Fq(.f.e.\))44 b(p)s(ermits)31 b(us)i(to)f(generalize)f(the)i(triplet)e(functional)f (equation)i(\(1.2\))328 4282 y(to)45 b(the)h(whole)f(class)h(\012)1242 4297 y Fe(0)1282 4282 y Fq(.)82 b(In)45 b(this)h(sense)h(the)f(idea)e (of)h(using)h(suc)m(h)g(a)g(generaliza-)328 4402 y(tion)33 b(of)h(\(1.2\))g(is)g(v)m(ery)i(close)e(to)g(the)h Fr(Grothendiec)m (k's)f Fq(magni\014cen)m(t)g(idea)f(of)h Fr(etale)328 4523 y(cohomologies)k Fq(on)h(the)h(category)g(of)g(algebraic)e(v)-5 b(arieties)38 b(\(although)h(our)g(idea)g(is)328 4643 y(ob)m(viously)47 b(not)g(so)g(magni\014cen)m(t!\).)87 b(Grothendiec)m(k)47 b(lo)s(ok)m(ed)g(at)g(the)g(notion)f(of)h(an)328 4763 y(op)s(en)39 b(set)g(of)g(a)f(top)s(ological)d(space)40 b(as)f(a)g(morphism)e(of)h(inclusion.)61 b(In)39 b(suc)m(h)h(a)f(w)m(a) m(y)328 4884 y(he)e(extended)i(the)e(notion)e(of)h(top)s(ology)f(from)h (a)g(set)h(in)m(to)f(a)g(category)-8 b(.In)37 b(particular)1898 5214 y(16)p eop %%Page: 17 17 17 16 bop 328 631 a Fq(Grothendiec)m(k)33 b(top)s(ologies)d(satisfy)j (all)d(axioms)h(of)h(set)h(top)s(ologies)e(in)g(the)i("cohomo-)328 751 y(logical)i(sense".)62 b(More)39 b(exactly)g(,)g(the)g(idea)f(of)f (etale)h(cohomologies)e(is)i(based)h(on)f(a)328 872 y(b)s(eautiful)24 b(notion)h(of)g("top)s(ology",)g(whic)m(h)i(di\013ers)e(from)g(a)g(top) s(ological)e(space)j(in,)h(that)328 992 y("op)s(en)35 b(sets")h(are)f(not)f(con)m(tained)i(in)e(a)g(set)i(,)g(but)f(satisfy)g (fundamen)m(tal)f(prop)s(erties,)328 1112 y(whic)m(h)f(mak)m(es)g(p)s (ossible)f(the)h(construction)g(of)f(a)g(necessary)j(cohomology)c (theory)-8 b(.)474 1233 y(Most)31 b(surprising)f(is)g(the)h(role)e(of)h (\014xed)i(p)s(oin)m(ts)e(in)g(the)g(general)g(categorical)f(sense)328 1353 y(in)41 b(the)h(case)h(of)e(general)h(Riemann)e(h)m(yp)s(othesis)j 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y Fq(\()p Fo(x)p Fq(\))p Fo(x)2043 3039 y Fk(s)p Fl(\000)p Fe(1)2170 3080 y Fo(dx)p Fq(;)45 b Fo(u)27 b Fq(=)h Fo(R)q(e)p Fq(\()p Fo(s)p Fq(\))f Fo(>)h Fq(0)p Fo(:)328 3281 y Fq(If)k Fo(n)h Fq(is)f(a)h(p)s(ositiv)m (e)f(in)m(teger,)g(w)m(e)i(ha)m(v)m(e,)g(on)e(substituting)g Fo(nx)h Fq(for)f Fo(x)p Fq(,)552 3525 y(\000\()p Fo(F)714 3540 y Fk(\025)760 3525 y Fq(\)\()p Fo(s)p Fq(\))27 b(=)1051 3408 y Ff(Z)1134 3434 y Fl(1)1097 3596 y Fe(0)1225 3525 y Fo(F)1288 3540 y Fk(\025)1333 3525 y Fq(\()p Fo(nx)p Fq(\)\()p Fo(nx)p Fq(\))1711 3484 y Fk(s)p Fl(\000)p Fe(1)1839 3525 y Fo(ndx)h Fq(=)g Fo(n)2193 3484 y Fk(s)2247 3408 y Ff(Z)2330 3434 y Fl(1)2293 3596 y Fe(0)2421 3525 y Fo(F)2484 3540 y Fk(\025)2529 3525 y Fq(\()p Fo(nx)p Fq(\))p Fo(x)2773 3484 y Fk(s)p Fl(\000)p Fe(1)2901 3525 y Fo(dx;)45 b(u)27 b(>)h Fq(0)p Fo(;)328 3741 y Fq(or)1150 3824 y(\000\()p Fo(F)1312 3839 y Fk(\025)1358 3824 y Fq(\)\()p Fo(s)p Fq(\))p 1150 3868 368 4 v 1286 3960 a Fo(n)1344 3931 y Fk(s)1583 3892 y Fq(=)1714 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Fo(omeg)t(a)p Fq(\()p Fo(nx)p Fq(\))p Fo(x)1350 4850 y Fk(s)p Fl(\000)p Fe(1)1505 4891 y Fn(j)h Fo(dx)f Fq(=)1829 4784 y Fl(1)1805 4808 y Ff(X)1798 4990 y Fk(n)p Fe(=1)1948 4774 y Ff(Z)2031 4801 y Fl(1)1994 4963 y Fe(0)2133 4891 y Fn(j)g Fo(F)2251 4906 y Fk(\025)2297 4891 y Fq(\()p Fo(nx)p Fq(\))h Fn(j)f Fo(x)2624 4850 y Fk(u)p Fl(\000)p Fe(1)2760 4891 y Fo(dx)h Fq(=)f(\000\()p Fn(j)h Fo(F)3215 4906 y Fk(\025)3288 4891 y Fn(j)p Fq(\)\()p Fo(u)p Fq(\))p Fo(\020)8 b Fq(\()p Fo(u)p Fq(\))1898 5214 y(17)p eop %%Page: 18 18 18 17 bop 328 631 a Fq(is)35 b(con)m(v)m(ergen)m(t)j(for)d Fo(u)g Fq(with)h Fo(u)c(>)h Fq(1,)j(w)m(e)h(can)f(in)m(terc)m(hange)h (the)f(order)g(of)f(summation)328 751 y(and)e(in)m(tegration,)e (obtaining)1199 995 y(\000\()p Fo(F)1361 1010 y Fk(\025)1406 995 y Fq(\)\()p Fo(s)p Fq(\))p Fo(\020)8 b Fq(\()p Fo(s)p Fq(\))54 b(=)1924 878 y Ff(Z)2007 904 y Fl(1)1970 1066 y Fe(0)2099 995 y Fo(\022)s Fq(\()p Fo(F)2248 1010 y Fk(\025)2293 995 y Fq(\)\()p Fo(x)p Fq(\))p Fo(x)2517 954 y Fk(s)p Fl(\000)p Fe(1)2645 995 y Fo(dx:)328 1246 y Fq(According)32 b(to)g(\()p Fo(\022)s(:f)5 b(:e:)p Fq(\),)34 b(this)e(can)h(b)s(e)g(rewritten)g(as)663 1509 y(\000\()p Fo(F)825 1524 y Fk(\025)871 1509 y Fq(\)\()p Fo(s)p Fq(\))p Fo(\020)8 b Fq(\()p Fo(s)p Fq(\))54 b(=)1389 1392 y Ff(Z)1472 1418 y Fe(1)1435 1580 y(0)1528 1509 y Fo(x)1583 1468 y Fk(s)p Fl(\000)p Fe(1)1710 1509 y Fo(\022)s Fq(\()p Fo(F)1859 1524 y Fk(\025)1905 1509 y Fq(\)\()p Fo(x)p Fq(\))p Fo(dx)22 b Fq(+)2300 1392 y Ff(Z)2383 1418 y Fl(1)2346 1580 y Fe(1)2475 1509 y Fo(x)2530 1468 y Fk(s)p Fl(\000)p Fe(1)2657 1509 y Fo(\022)s Fq(\()p Fo(F)2806 1524 y Fk(\025)2852 1509 y Fq(\)\()p Fo(x)p Fq(\))p Fo(dx)28 b Fq(=)477 1853 y(=)581 1736 y Ff(Z)664 1762 y Fe(1)627 1925 y(0)720 1853 y Fo(x)775 1812 y Fk(s)p Fl(\000)p Fe(1)902 1853 y Fq(\()p Fo(x)995 1812 y Fl(\000)p Fe(1)1090 1853 y Fo(\022)s Fq(\()p Fo(F)1239 1868 y Fk(\025)1285 1853 y Fq(\)\()1374 1786 y(1)p 1371 1830 56 4 v 1371 1922 a Fo(x)1436 1853 y Fq(\))22 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(an)g(imaginary)d(part)k(of)f(a)g(complex)f(n)m(um)m(b)s(er.)65 b(W)-8 b(e)39 b(consider)1898 5214 y(18)p eop %%Page: 19 19 19 18 bop 328 631 a Fr(amplitudes)32 b Fo(A)d Fq(:)g Fm(R)38 b Fn(\000)-16 b(!)28 b Fm(C)c Fq(,)33 b(whic)m(h)h(are)g Fn(F)d(\000)23 b Fr(admissible)31 b Fq(and)j Fr(an)m(t)m(ysymmetric)328 751 y Fq(\(i.e.)43 b Fo(A)p Fq(\()p Fn(\000)p Fo(x)p Fq(\))28 b(=)g Fo(A)p Fq(\()p Fo(x)p Fq(\)\).)44 b(Therefore)34 b(they)f(satisfy)g(the)g(functional)e(equation:)1534 938 y Fn(F)10 b Fq(\()p Fo(A)p Fq(\))55 b(=)g(2)p Fo(I)8 b(m)p Fn(F)2208 953 y Fe(+)2267 938 y Fq(\()p Fo(A)p Fq(\))p Fo(:)874 b Fq(\(3.41\))474 1125 y(In)32 b(the)g(sequel)g(w)m(e) h(consider)f(only)f Fr(Con)m(tin)m(uous)p Fq(,)g(\(Leb)s(esgue\)-)p Fr(In)m(tegrable)h Fq(\(on)328 1245 y Fm(R)9 b Fq(\))36 b(and)h(\(strictly\)-)p Fr(Decreasing)k(amplitudes)35 b Fq(\(CID-amplitudes)g(in)h(short\))h Fo(A)e Fq(=)328 1365 y Fo(A)p Fq(\()p Fo(x)p Fq(\).)58 b(Moreo)m(v)m(er,)40 b(a)d(w)m(ord)h("amplitude")c(alw)m(a)m(ys)k(in)f(this)g(pap)s(er)g (mean,)h(that)f Fo(A)g Fq(is)328 1486 y Fr(p)s(ositiv)m(e)p Fq(.)62 b(With)39 b(suc)m(h)h(an)f(amplitude)f Fo(A)h Fq(is)g(connected)h(a)f Fr("dissapp)s(ear)46 b(\(fade\))328 1606 y(w)m(a)m(v)m(e")33 b Fq(:)43 b Fo(A)p Fq(\()p Fo(x)p Fq(\))p Fo(sin)p Fq(\()p Fo(ax)p Fq(\))34 b(\(with)e(a)h("phase")g Fo(a)p Fq(\))f(and)h Fn(F)2376 1621 y Fe(+)2435 1606 y Fr(-v)m(olume)1176 1837 y Fn(F)1248 1852 y Fe(+)1306 1837 y Fq(\()p Fo(AsinL)1620 1852 y Fk(a)1663 1837 y Fq(\))55 b(:=)1914 1720 y Ff(Z)1997 1746 y Fe(+)p Fl(1)1960 1908 y Fe(0)2144 1837 y Fo(A)p Fq(\()p Fo(x)p Fq(\))p Fo(sin)p Fq(\()p Fo(ax)p Fq(\))p Fo(dx:)517 b Fq(\(3.42\))328 2057 y(Here)34 b Fo(L)625 2072 y Fk(a)667 2057 y Fq(\()p Fo(u)p Fq(\))29 b(:=)g Fo(au;)17 b(u)29 b Fn(2)h Fm(R)42 b Fq(stands)34 b(for)f(the)h Fr(linear)k(function)p Fq(.)46 b(The)35 b(b)s(elo)m(w)e(lemma)328 2178 y(is)27 b("in)m(tuitionally)d (ob)m(vious",)29 b(but)f(w)m(e)h(giv)m(e)f(its)g(pro)s(of)f(for)g(the)h (sak)m(e)i(of)d(completness.It)328 2298 y(also)j(underline)g(some)h(sp) s(ecial)f(place)g(of)h(\(CID\)-amplitudes)e(in)h(the)h(harmonic)e (anal-)328 2419 y(ysis.)328 2593 y Fr(Lemma)37 b(3)49 b Fb(F)-7 b(or)34 b(e)-5 b(ach)34 b(CID-amplitude)f Fo(A)28 b Fq(:)g Fm(R)36 b Fn(\000)-16 b(!)27 b Fm(R)2398 2557 y Fl(\003)2398 2618 y Fe(+)2492 2593 y Fb(and)34 b(phase)g Fo(a)28 b Fn(2)g Fm(R)3189 2557 y Fl(\003)3263 2593 y Fb(holds)1595 2780 y Fn(F)1667 2795 y Fe(+)1725 2780 y Fq(\()p Fo(AsinL)2039 2795 y Fk(a)2082 2780 y Fq(\))55 b Fo(>)g Fq(0)p Fo(:)935 b Fq(\(3.43\))328 2967 y Fr(Pro)s(of)p Fq(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)328 3198 y Fn(F)400 3213 y Fe(+)459 3198 y Fq(\()p Fo(AsinL)773 3213 y Fk(a)815 3198 y Fq(\))55 b(=)1040 3081 y Ff(Z)1123 3107 y Fe(+)p Fl(1)1086 3269 y Fe(0)1269 3198 y Fo(A)p Fq(\()p Fo(x)p Fq(\))p Fo(sin)p Fq(\()p Fo(ax)p Fq(\))p Fo(dx)i Fq(=)2117 3090 y Fl(1)2092 3115 y Ff(X)2086 3296 y Fk(n)p Fe(=0)2236 3081 y Ff(Z)2319 3107 y Fe(\()p Fk(n)p Fe(+1\))p Fk(\031)r(=a)2282 3269 y(n\031)r(=a)2643 3198 y Fo(A)p Fq(\()p Fo(x)p Fq(\))p Fo(sin)p Fq(\()p Fo(ax)p Fq(\))p Fo(dx)g Fq(=)e(\()3502 3090 y Fe(+)p Fl(1)3505 3115 y Ff(X)3498 3296 y Fk(n)p Fe(=0)3631 3198 y Fq(\()p Fn(\000)p Fq(1\))3833 3157 y Fk(n)3880 3198 y Fo(A)p Fq(\()p Fo(x)4046 3213 y Fk(n)4093 3198 y Fq(\)\))p Fo(P)14 b Fq(\()p Fo(a)p Fq(\))p Fo(;)328 3479 y Fq(where)29 b Fo(P)14 b Fq(\()p Fo(a)p Fq(\))27 b(=)939 3408 y Ff(R)995 3429 y Fk(\031)r(=a)978 3501 y Fe(0)1131 3479 y Fo(sin)p Fq(\()p Fo(ax)p Fq(\))p Fo(dx)i(>)e Fq(0)g(and)h(a)f(sequence)j Fn(f)p Fo(x)2529 3494 y Fk(n)2576 3479 y Fn(g)d Fq(with)g Fo(x)2925 3494 y Fk(n)3000 3479 y Fn(2)h Fq([)p Fo(n\031)t(=a;)17 b Fq(\()p Fo(n)11 b Fq(+)328 3600 y(1\))p Fo(\031)t(=a)p Fq(])41 b(is)f(determined)h (according)f(to)h(the)g Fr(mean)47 b(v)-6 b(alue)47 b(theorem)p Fq(.)68 b(Since)41 b Fo(A)g Fq(is)328 3720 y Fr(p)s(ositiv)m(e,)j(con)m (tin)m(uous,)h(in)m(tegrable)37 b Fq(and)h Fr(decreasing)p Fq(\(strictly\),)h(then)f(w)m(e)h(ob-)328 3841 y(tain)31 b(the)i(thesis)h(of)e(the)h(lemma.)474 3961 y(W)-8 b(e)30 b(sa)m(y)g(that)f(a)g(quasi-\014xed)h(p)s(oin)m(t)e Fo(F)1889 3976 y Fk(\025)1962 3961 y Fq(=)g(\()p Fo(F)2181 3925 y Fk(A)2167 3986 y(\025)2237 3961 y Fq(\))g Fn(2)g Fo(Q)p Fq(\012)i(is)f(a)g Fr(\(RH\)-quasi-\014xed)328 4081 y(p)s(oin)m(t)37 b(of)g Fn(F)42 b Fq(i\013)32 b(there)h(exists)g(suc)m(h)h(a)f (\(CID\)-amplitude)d Fo(A)e Fq(=)f Fo(A)p Fq(\()p Fo(x)p Fq(\);)17 b Fo(x)29 b(>)e Fq(0)32 b(that)869 4268 y(\()p Fo(R)q(H)r(:Q:f)5 b(:)p Fq(\))56 b(\()p Fo(F)1485 4227 y Fk(A)1471 4293 y(\025)1592 4268 y Fn(\000)50 b Fo(G)p Fq(\)\()p Fo(x)p Fq(\))83 b(=)g Fo(A)p Fq(\()p Fo(x)p Fq(\))28 b Fo(f)11 b(or)30 b(al)r(l)h(x)d Fn(\025)g Fq(1)p Fo(:)292 b Fq(\(3.44\))474 4455 y(The)38 b(problem)d(of)g(the)i (existence)h(of)e(\(RH\)-\014xed)h(p)s(oin)m(ts)f(immediately)d(leads)j (to)328 4575 y(the)c(problem)e(of)g(the)i(solution)e(of)h(the)g Fr(second)37 b(order)31 b Fq(\(non-homogeneous\))g Fr(F)-9 b(red-)328 4695 y(holm)36 b(in)m(tegral)g(equation)d Fq(of)f(the)h(form)807 4926 y(\()p Fo(F)14 b(E)6 b Fq(\))55 b Fo(f)11 b Fq(\()p Fo(x)p Fq(\))27 b(=)h Fo(\025)1488 4809 y Ff(Z)1570 4836 y Fk(b)1533 4998 y(a)1621 4926 y Fo(K)7 b Fq(\()p Fo(x;)17 b(y)t Fq(\))p Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)56 b Fq(=)g Fo(g)t Fq(\()p Fo(x)p Fq(\))f(;)17 b Fo(x)27 b Fn(2)i Fq([)p Fo(a;)17 b(b)p Fq(])p Fo(;)1898 5214 y Fq(19)p eop %%Page: 20 20 20 19 bop 328 631 a Fq(with)32 b(a)g Fr(symmetric)j(singular)p Fq(\(i.e.)43 b(on)32 b Fm(R)1975 646 y Fe(+)2034 631 y Fq(\))g(and)h Fr(F)-9 b(o)m(x)38 b(k)m(ernel)p Fq(\(i.e.\))1068 833 y(\()p Fo(F)14 b(K)7 b Fq(\))138 b Fo(K)7 b Fq(\()p Fo(x;)17 b(y)t Fq(\))55 b(=)g Fo(K)7 b Fq(\()p Fo(xy)t Fq(\))55 b(\()p Fo(or)30 b(K)7 b Fq(\()p Fo(x=y)t Fq(\)\))p Fo(:)474 1034 y Fq(The)42 b(F)-8 b(redholm)39 b(in)m(tegral)g (equations)i(whic)m(h)g(k)m(ernels)h(satis\014es)f(\(FK\)-condition)328 1154 y(are)c(called)g(the)h Fr(F)-9 b(o)m(x)43 b(in)m(tegral)e (equations)d Fq(\(cf.e.g.[KKM,)g(Sect.I)s(I.23]\))g(in)f(opp)s(o-)328 1275 y(siteto)h(the)h(more)e(p)s(opular)h Fr(Wiener-Hopf)44 b(in)m(tegral)e(equations)d Fq(whic)m(h)g(k)m(ernels)328 1395 y(satis\014es)33 b("arithmetically)c(dual")j(condition)f(to)h(the) h(\(FK\):)1294 1597 y(\()p Fo(W)14 b(H)8 b Fq(\))110 b Fo(K)7 b Fq(\()p Fo(x;)17 b(y)t Fq(\))55 b(=)g Fo(K)7 b Fq(\()p Fo(x)23 b Fn(\000)f Fo(y)t Fq(\))p Fo(:)328 1798 y Fr(Remark)37 b(3)49 b Fb(It)35 b(is)g(very)h(surprising)f(that)h (the)g(the)-5 b(ory)35 b(of)h(inte)-5 b(gr)g(al)35 b(e)-5 b(quations)35 b(-which)328 1919 y(c)-5 b(omp)g(aring)30 b(with)h(the)g(the)-5 b(ory)32 b(of)f(p)-5 b(artial)31 b(di\013er)-5 b(ential)31 b(e)-5 b(quations)30 b(-)i(is)f(not)g(such)g (p)-5 b(opu-)328 2039 y(lar.)41 b(\(The)25 b(the)-5 b(ory)26 b(of)f(inte)-5 b(gr)g(al)25 b(e)-5 b(quations)25 b(se)-5 b(ems)25 b(to)g(b)-5 b(e)25 b(very)h("mo)-5 b(dest")25 b(if)g(we)g(c)-5 b(omp)g(ar)g(e)328 2159 y(it)35 b(with)g(the)g(the)-5 b(ory)35 b(of)f(di\013er)-5 b(ential)34 b(e)-5 b(quations,)35 b(but)g(in)g(this)f(c)-5 b(ase)35 b(it)g(is)f(de)-5 b(ciding\).)474 2280 y(Observe)38 b(that)h(ac)-5 b(c)g(or)g(ding)37 b(to)i(the)f (formal)g(r)-5 b(esemblense)37 b(of)h(a)h(F)-7 b(r)i(e)g(dholm)36 b(inte)-5 b(gr)g(al)328 2400 y(e)g(quation)34 b(of)h(the)g(se)-5 b(c)g(ond)34 b(or)-5 b(der)1161 2646 y Fo(\025)1235 2529 y Ff(Z)1317 2555 y Fk(b)1280 2718 y(a)1368 2646 y Fo(K)7 b Fq(\()p Fo(x;)17 b(y)t Fq(\))p Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)51 b Fq(+)f Fo(f)11 b Fq(\()p Fo(x)p Fq(\))55 b(=)g Fo(g)t Fq(\()p Fo(x)p Fq(\))p Fo(;)328 2876 y Fb(to)35 b(a)g(non-homo)-5 b(gene)g(ous)32 b(di\013er)-5 b(ential)34 b(e)-5 b(quation)35 b(of)f(the)h(\014rst)g(or)-5 b(der)1338 3078 y Fo(\025)p Fq(\()p Fo(x)p Fq(\))p Fo(f)1585 3037 y Fl(0)1608 3078 y Fq(\()p Fo(x)p Fq(\))106 b(+)77 b Fo(f)11 b Fq(\()p Fo(x)p Fq(\))28 b(=)55 b Fo(g)t Fq(\()p Fo(x)p Fq(\))p Fo(;)328 3279 y Fb(the)45 b(mo)-5 b(dern)45 b(name)f(should)h(b)-5 b(e)45 b(a)g(")g(non-homo)-5 b(gene)g(ous)43 b(inte)-5 b(gr)g(al)45 b(e)-5 b(quation)45 b(of)g(the)328 3400 y(\014rst)f(or)-5 b(der)43 b(\(inste)-5 b(ad)43 b(a)g(historic)-5 b(al)43 b(one\).)70 b(Obviously)43 b(-)h(b)-5 b(eside)42 b(the)i(same)f Fr(formal)328 3520 y(structure)49 b Fb(of)g(ab)-5 b(ove)49 b(inte)-5 b(gr)g(al)49 b(and)g(di\013er)-5 b(ential)49 b(e)-5 b(quation)49 b(-)h(the)f Fr(structure)56 b(of)328 3641 y(solutions)41 b Fb(is)h(quite)h (di\013er)-5 b(ent.)66 b(F)-7 b(or)42 b(example)f(-)h(inte)-5 b(gr)g(al)42 b(op)-5 b(er)g(ator)g(ors)42 b(ar)-5 b(e)42 b(glob)-5 b(al,)328 3761 y(wher)g(e)g(as)34 b(di\013er)-5 b(ential)34 b(op)-5 b(er)g(ators)34 b(have)g(got)h(a)g(total)5 b(ly)35 b(lo)-5 b(c)g(al)34 b(structur)-5 b(e.)46 b(Pr)-5 b(ob)g(ably)35 b(the)328 3881 y(most)45 b(imp)-5 b(ortantant)44 b(examples)g(of)g(inte)-5 b(gr)g(al)45 b(op)-5 b(er)g(ators)44 b(ar)-5 b(e)45 b(the)g(F)-7 b(ourier,)47 b(L)-5 b(aplac)g(e)328 4002 y(and)34 b(Mel)5 b(lin)35 b(tr)-5 b(ansforms.)474 4122 y(The)25 b(F)-7 b(ox)23 b(inte)-5 b(gr)g(al)25 b(e)-5 b(quations)24 b(c)-5 b(an)25 b(b)-5 b(e)24 b(solve)-5 b(d)24 b Fr(algebraically)f Fb(by)i(using)g(the)g Fr(Mellin)328 4242 y(transform)p Fb(.)87 b(It)49 b(is)g(very)g(detaile)-5 b(d)48 b(explaine)-5 b(d)48 b(in)h([KKM])g(,)g(what)g(is)f(a)h(little)h (bit)328 4363 y(amazing)31 b(c)-5 b(oincidenc)g(e,)31 b(sinc)-5 b(e)31 b(in)h([KKM,)g(Se)-5 b(ct.II.23])30 b(is)i(exactly)g(solve)-5 b(d)31 b(such)h(a)g(F)-7 b(ox)328 4483 y(e)i(quation)36 b(what)g(we)g(ne)-5 b(e)g(d.)48 b(The)35 b Fr(Mellin)i(transform)f Fb(is)g(a)g(sp)-5 b(e)g(ci\014c)35 b(functor,)i(which)328 4604 y(r)-5 b(e)g(duc)g(es)36 b(a)g(r)-5 b(ather)37 b(di\016cult)f(analytic)g(pr)-5 b(oblem)36 b(of)g(the)g(solution)g(of)h(the)f(inte)-5 b(gr)g(al)36 b(F)-7 b(ox)328 4724 y(e)i(quation)40 b(to)h(a)g(r)-5 b(elatively)40 b(e)-5 b(asy)41 b Fr(algebraic)f Fb(pr)-5 b(oblem)40 b(of)g(the)h(solution)g(of)f(an)g(op)-5 b(er-)328 4844 y(ator)45 b(e)-5 b(quation.)76 b(\(It)45 b(is)g(wel)5 b(l-known)43 b(that)j(in)f(the)g(the)-5 b(ory)46 b(of)f(or)-5 b(dinary)45 b(line)-5 b(ar)44 b(dif-)328 4965 y(fer)-5 b(ential)36 b(e)-5 b(quations)37 b(of)g(a)f(\014nite)h(or)-5 b(der,)37 b(the)g(same)f(r)-5 b(ole)37 b(is)g(playe)-5 b(d)36 b(by)h(the)g Fr(Laplace)1898 5214 y Fq(20)p eop %%Page: 21 21 21 20 bop 328 631 a Fr(transform)p Fb(\).)50 b(Obviously,)37 b(the)f(ful)5 b(l)37 b(algebr)-5 b(aization)35 b(of)i(a)g(F)-7 b(ox)35 b(e)-5 b(quation)36 b(is)h(p)-5 b(ossible)328 751 y(ac)g(c)g(or)g(ding)42 b(to)h(\()g(a)g(de)-5 b(ep)g(er)42 b(pr)-5 b(op)g(erty)43 b(of)g(Mel)5 b(lin)42 b(tr)-5 b(ansform\):)61 b(the)43 b(the)-5 b(or)g(em)42 b(on)h(the)328 872 y Fr(con)m(v)m(olution)33 b Fb(for)i(Mel)5 b(lin)35 b(tr)-5 b(ansform:)1055 1133 y Fo(M)10 b Fq([)1186 1016 y Ff(Z)1269 1043 y Fe(+)p Fl(1)1233 1205 y Fe(0)1416 1133 y Fo(f)h Fq(\()p Fo(t)p Fq(\))p Fo(\036)p Fq(\()1692 1066 y Fo(x)p 1692 1110 56 4 v 1702 1202 a(t)1757 1133 y Fq(\))1805 1066 y Fo(dt)p 1805 1110 86 4 v 1830 1202 a(t)1900 1133 y Fq(])56 b(=)f([\()p Fo(M)10 b(f)h Fq(\))23 b Fn(\001)e Fq(\()p Fo(M)10 b(\036)p Fq(\)]\()p Fo(s)p Fq(\))p Fo(:)451 b Fq(\(3.45\))328 1385 y Fb(Final)5 b(ly,)35 b(let's)g(r)-5 b(emark)35 b(that)h(the)g(R)n(iemann)e(hyp)-5 b(othesis)35 b(\(RH\))h(is)f(then)g(very)h(strictly)328 1506 y(asso)-5 b(ciate)g(d)29 b(with)h(the)g(pr)-5 b(oblem)30 b(of)g(the)g(solution)g(of)g(the)g(F)-7 b(ox)29 b(e)-5 b(quation.)43 b(\(Blisko)29 b(pr)-5 b(acy)328 1626 y(\(Calkowalnosc)33 b(to)i(\(RH\)?\)\).)474 1828 y Fq(W)-8 b(e)33 b(are)g(ready)g(no)m(w)g (to)g(pro)m(v)m(e)g(a)g(follo)m(wing)d(preliminary)g(tec)m(hnical)i (result)g(:)328 2029 y Fr(Prop)s(osition)j(1)49 b Fb(\()p Fr(On)59 b(the)g(existence)f(of)h(\(RH\)-quasi-\014xed)g(p)s(oin)m(ts)p Fb(\))51 b(F)-7 b(or)328 2150 y(e)i(ach)44 b(p)-5 b(ositive)44 b(CID-amplitude)f Fo(A)i Fq(=)h Fo(A)p Fq(\()p Fo(x)p Fq(\))f Fb(fr)-5 b(om)44 b Fn(S)7 b Fq(\()p Fm(R)j Fq(\))44 b Fb(and)g(a)h(sp)-5 b(e)g(ctr)g(al)44 b(p)-5 b(ar)g(ame-)328 2270 y(ter)35 b Fo(\025)27 b Fn(6)p Fq(=)h Fn(\006)p Fq(1)35 b Fb(ther)-5 b(e)34 b(exists)g Fr(\(RH\)-quasi-\014xed)j(p)s (oin)m(t)d Fo(F)2547 2285 y Fk(\025)2627 2270 y Fb(asso)-5 b(ciate)g(d)33 b(with)i Fo(A)p Fb(,)f(i.e.)1251 2488 y Fq(\()p Fo(F)1352 2503 y Fk(\025)1447 2488 y Fn(\000)50 b Fo(\013)q(G)p Fq(\()p Fo(x)p Fq(\)\))55 b(=)h Fo(A)p Fq(\()p Fn(j)27 b Fo(x)h Fn(j)p Fq(\);)44 b Fo(x)28 b Fn(2)g Fm(R)9 b Fo(:)591 b Fq(\(3.46\))328 2706 y Fb(Mor)-5 b(e)g(over)35 b Fo(F)820 2721 y Fk(\025)893 2706 y Fn(2)28 b(S)7 b Fq(\()p Fm(R)i Fq(\))p Fb(.)328 2907 y Fr(Pro)s(of)p Fq(.)46 b(W)-8 b(e)34 b(searc)m(h)h Fo(F)1202 2922 y Fk(\025)1281 2907 y Fq(in)d(the)i(form)f(:)45 b Fo(F)1931 2922 y Fk(\025)1976 2907 y Fq(\()p Fo(x)p Fq(\))30 b(=)f Fo(f)11 b Fq(\()p Fo(x)p Fq(\))23 b(+)f Fo(\025)2632 2881 y Fq(^)2610 2907 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))34 b(,)g(for)f(an)g(unkno)m(wn)328 3027 y(and)g(symmetric)e Fo(f)39 b Fq(:)28 b Fm(R)36 b Fn(\000)-16 b(!)27 b Fm(R)9 b Fq(.)474 3148 y(According)38 b(symmetricit)m(y)f(of)g Fo(f)11 b Fq(,)39 b(w)m(e)g(see)g(that)f(the)g(equalit)m(y)f(\(3.)59 b(\))g(reduces)40 b(to)328 3268 y(the)d(problem)f(of)h(the)g(solution)f (of)h(the)g(follo)m(wing)d Fr(F)-9 b(o)m(x)43 b(in)m(tegral)e(equation) c Fq(\(with)328 3389 y(the)c(k)m(ernel)g Fo(K)7 b Fq(\()p Fo(xy)t Fq(\))27 b(:=)h Fo(cos)p Fq(\(2)p Fo(\031)t(xy)t Fq(\)\))j(:)448 3639 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))49 b(+)h(2)p Fo(\025)936 3522 y Ff(Z)1018 3548 y Fl(1)981 3710 y Fe(0)1110 3639 y Fo(cos)p Fq(\(2)p Fo(\031)t(xy)t Fq(\))p Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)56 b Fq(=)f Fo(A)p Fq(\()p Fo(x)p Fq(\))22 b(+)g Fo(\013)q(G)p Fq(\()p Fo(x)p Fq(\)\(=:)28 b Fo(g)2820 3654 y Fk(A)2877 3639 y Fq(\()p Fo(x)p Fq(\)\))55 b(;)17 b Fo(x)28 b Fn(2)g Fm(R)3388 3654 y Fe(+)3447 3639 y Fo(:)3317 3793 y Fq(\(3.47\))328 3914 y(Multiplying)i(b)s(oth)j(sides)h(of)e(\(.\))45 b Fo(x)1639 3877 y Fk(s)p Fl(\000)p Fe(1)1799 3914 y Fq(,)33 b(in)m(tegrating)e(w.r.t.)46 b(the)33 b(Leb)s(esgue)h(measure) 328 4034 y Fo(dx)f Fq(on)f Fm(R)668 4049 y Fe(+)759 4034 y Fq(and)h(applying)e Fr(F)-9 b(ubini)37 b(theorem)32 b Fq(w)m(e)h(get)385 4164 y Ff(Z)468 4190 y Fl(1)431 4353 y Fe(0)559 4281 y Fo(g)606 4296 y Fk(A)663 4281 y Fq(\()p Fo(x)p Fq(\))p Fo(x)849 4240 y Fk(s)p Fl(\000)p Fe(1)976 4281 y Fo(dx)56 b Fq(=)1269 4164 y Ff(Z)1352 4190 y Fl(1)1315 4353 y Fe(0)1443 4281 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))p Fo(x)1688 4240 y Fk(s)p Fl(\000)p Fe(1)1816 4281 y Fo(dx)22 b Fq(+)g Fo(\025)2116 4164 y Ff(Z)2199 4190 y Fl(1)2162 4353 y Fe(0)2290 4281 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)2594 4164 y Ff(Z)2677 4190 y Fl(1)2640 4353 y Fe(0)2769 4281 y Fo(cos)p Fq(\(2)p Fo(\031)t(xy)t Fq(\))p Fo(x)3250 4240 y Fk(s)p Fl(\000)p Fe(1)3376 4281 y Fo(dx:)3317 4436 y Fq(\(3.48\))328 4556 y(Denoting)29 b(b)m(y)907 4531 y(~)880 4556 y Fo(K)8 b Fq(\()p Fo(s)p Fq(\))30 b(the)h Fr(Mellin)i(transform)d Fq(of)g Fo(K)7 b Fq(\()p Fo(x)p Fq(\))28 b(=)f Fo(cos)p Fq(\(2)p Fo(\031)t(x)p Fq(\))j(and)h(applying)328 4676 y Fr(con)m(v)m(olution)36 b(theorem)h(for)g Fo(M)43 b Fq(\(.\))h(,)32 b(w)m(e)i(can)f(write)f (\(.\))43 b(of)32 b(the)h(form)980 4926 y(\()p Fo(M)10 b(g)1169 4941 y Fk(A)1226 4926 y Fq(\)\()p Fo(s)p Fq(\))55 b(=)g(\()p Fo(M)10 b(f)h Fq(\)\()p Fo(s)p Fq(\))50 b(+)2135 4901 y(~)2109 4926 y Fo(K)7 b Fq(\()p Fo(s)p Fq(\))2338 4809 y Ff(Z)2421 4836 y Fl(1)2384 4998 y Fe(0)2512 4926 y Fo(f)k Fq(\()p Fo(y)t Fq(\))p Fo(y)2751 4885 y Fl(\000)p Fk(s)2841 4926 y Fo(dy)t(:)346 b Fq(\(3.49\))1898 5214 y(21)p eop %%Page: 22 22 22 21 bop 328 631 a Fq(It)30 b(is)f(easy)i(to)f(c)m(hec)m(k)i(that)1324 560 y Ff(R)1379 586 y Fl(1)1363 656 y Fe(0)1471 631 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(y)1710 595 y Fl(\000)p Fk(s)1799 631 y Fo(dy)31 b Fq(=)c(\()p Fo(M)10 b(f)h Fq(\)\(1)17 b Fn(\000)g Fo(s)p Fq(\).)43 b(Therefore)31 b(the)f(equalit)m(y)328 751 y(\(.\))43 b(w)m(e)34 b(can)f(write)f(of)g (the)h Fr(op)s(erator)38 b(form)p Fq(:)1024 971 y(\()p Fo(M)10 b(g)1213 986 y Fk(A)1271 971 y Fq(\)\()p Fo(s)p Fq(\))55 b(=)g(\()p Fo(M)10 b(f)h Fq(\)\()p Fo(s)p Fq(\))50 b(+)2180 946 y(~)2154 971 y Fo(K)7 b Fq(\()p Fo(s)p Fq(\)\()p Fo(M)j(f)h Fq(\)\(1)22 b Fn(\000)h Fo(s)p Fq(\))p Fo(:)392 b Fq(\(3.50\))328 1191 y(Replacing)25 b(in)i(\(.\))41 b Fo(s)27 b Fq(b)m(y)h(1)11 b Fn(\000)g Fo(s)26 b Fq(\(i.e.)41 b(acting)26 b(the)i(generator)e Fo(m)i Fq(of)e Fo(M)10 b(od)p Fq(\()p Fm(C)24 b Fq(\)\))j(w)m(e)h(obtain)383 1411 y(\()p Fo(M)10 b(g)572 1426 y Fk(A)630 1411 y Fq(\)\(1)22 b Fn(\000)g Fo(s)p Fq(\))55 b(=)h(\()p Fo(M)10 b(f)h Fq(\)\(1)22 b Fn(\000)g Fo(s)p Fq(\))50 b(+)1880 1386 y(~)1854 1411 y Fo(K)7 b Fq(\(1)22 b Fn(\000)g Fo(s)p Fq(\)\()p Fo(M)10 b(f)h Fq(\)\()p Fo(s)p Fq(\))p Fo(:)693 b Fq(\(3.51\))328 1631 y(F)-8 b(rom)31 b(\(.\))43 b(and)33 b(\(.\))43 b(follo)m(ws)32 b(that)952 1923 y(\()p Fo(M)10 b(f)h Fq(\)\()p Fo(s)p Fq(\))56 b(=)1510 1856 y(\()p Fo(M)10 b(g)1699 1871 y Fk(A)1756 1856 y Fq(\)\()p Fo(s)p Fq(\))22 b Fn(\000)h Fq(2)p Fo(\025)p Fq(\()p Fo(M)10 b(g)2333 1871 y Fk(A)2390 1856 y Fq(\)\(1)22 b Fn(\000)g Fo(s)p Fq(\))2746 1830 y(~)2720 1856 y Fo(K)7 b Fq(\()p Fo(s)p Fq(\))p 1510 1900 1423 4 v 1766 2001 a(1)22 b Fn(\000)h Fq(4)p Fo(\025)2043 1972 y Fe(2)2108 1976 y Fq(~)2082 2001 y Fo(K)7 b Fq(\()p Fo(s)p Fq(\))2320 1976 y(~)2294 2001 y Fo(K)g Fq(\(1)22 b Fn(\000)g Fo(s)p Fq(\))2942 1923 y Fo(:)348 b Fq(\(3.52\))328 2208 y(The)27 b(form)m(ula)d(\(.\))41 b(giv)m(es)26 b(an)g(op)s(erator)g(solution)e(of)i(the)g(F)-8 b(o)m(x)26 b(equation)g(\(.\).)41 b(According)328 2329 y(to)27 b(the)h(form)m(ula)d(concerning)j(the)g Fr(in)m(v)m(ersion)e Fq(of)h(the)h(Mellin)d(transform,)j(w)m(e)g(\014nd)g(the)328 2449 y(solution)j(of)h(the)h(F)-8 b(o)m(x)33 b(equation)f(\(.\))43 b(as)383 2741 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))56 b(=)816 2673 y(1)p 770 2718 141 4 v 770 2809 a(2)p Fo(\031)t(i)937 2624 y Ff(Z)1020 2650 y Fk(\033)r Fe(+)p Fk(i)p Fl(1)983 2812 y Fk(\033)r Fl(\000)p Fk(i)p Fl(1)1243 2673 y Fq(\()p Fo(M)10 b(g)1432 2688 y Fk(A)1490 2673 y Fq(\)\()p Fo(s)p Fq(\))22 b Fn(\000)g Fq(2)p Fo(\025)p Fq(\()p Fo(M)10 b(g)2066 2688 y Fk(A)2123 2673 y Fq(\)\(1)22 b Fn(\000)h Fo(s)p Fq(\))2480 2648 y(~)2454 2673 y Fo(K)7 b Fq(\()p Fo(s)p Fq(\))p 1243 2718 1423 4 v 1499 2819 a(1)22 b Fn(\000)h Fq(4)p Fo(\025)1776 2790 y Fe(2)1841 2794 y Fq(~)1815 2819 y Fo(K)7 b Fq(\()p Fo(s)p Fq(\))2053 2794 y(~)2027 2819 y Fo(K)g Fq(\(1)22 b Fn(\000)h Fo(s)p Fq(\))2676 2741 y Fo(x)2731 2700 y Fl(\000)p Fk(s)2823 2741 y Fo(ds:)370 b Fq(\(3.53\))474 3026 y(A)m(t)46 b(this)g(place)g(w)m(e)h(rep)s(eat)f (the)g(calculations)e(\(practically)g(without)h(c)m(hanges\))328 3147 y(giv)m(en)33 b(at)f(the)h(last)f(example)g(of)g(the)h(Section)f (I)s(I.23)h(from)e([KKM].)474 3267 y(W)-8 b(e)33 b(ha)m(v)m(e)1308 3362 y(~)1282 3388 y Fo(K)7 b Fq(\()p Fo(s)p Fq(\))56 b(=)f(2)p Fo(\025)1804 3270 y Ff(Z)1886 3297 y Fl(1)1849 3459 y Fe(0)1977 3388 y Fo(x)2032 3346 y Fk(s)p Fl(\000)p Fe(1)2160 3388 y Fo(cos)p Fq(\(2)p Fo(\031)t(x)p Fq(\))p Fo(dx:)650 b Fq(\(3.54\))328 3590 y(T)-8 b(o)24 b(calculate)f(the)i(ab) s(o)m(v)m(e)g(in)m(tegral)e(w)m(e)i(recall)e(the)i(\014rst)g(the)f (de\014nition)g(of)g(the)g(classical)328 3711 y Fr(Gamma)p Fq(:)1470 3714 y Ff(Z)1553 3740 y Fl(1)1516 3903 y Fe(0)1644 3831 y Fo(e)1689 3790 y Fl(\000)p Fk(x)1788 3831 y Fo(x)1843 3790 y Fk(z)s Fl(\000)p Fe(1)1973 3831 y Fo(dx)56 b Fq(=)f(\000\()p Fo(z)t Fq(\))p Fo(:)838 b Fq(\(3.55\))328 4040 y(If)37 b(in)g(\(.\))59 b(w)m(e)38 b(mak)m(e)g(a)g(turn)f(of)h(the)g(radius)f (of)g(in)m(tegration)f(at)h Fo(\031)t(=)p Fq(2)g(\()h(it)e(is)h(p)s (ossible)328 4160 y(according)i(to)g(the)h Fr(Jordan)46 b(lemma)38 b Fq(if)g Fo(z)44 b Fn(2)c Fq(\(0)p Fo(;)17 b Fq(1\))39 b(,)i(i.e.)64 b(w)m(e)40 b(formally)d(mak)m(e)i(the)328 4280 y(substitution)32 b(:)p Fo(x)c Fq(=)g(2)p Fo(\031)t(y)t(i)p Fq(\))j(w)m(e)j(obtain)d(the)i(form)m(ula:)1245 4416 y Ff(Z)1328 4442 y Fl(1)1291 4604 y Fe(0)1419 4533 y Fo(e)1464 4492 y Fl(\000)p Fe(2)p Fk(\031)r(iy)1663 4533 y Fq(\(2)p Fo(\031)t(iy)t Fq(\))1932 4492 y Fk(z)s Fl(\000)p Fe(1)2061 4533 y Fq(2)p Fo(\031)t(idy)58 b Fq(=)d(\000\()p Fo(z)t Fq(\))p Fo(;)614 b Fq(\(3.56\))328 4753 y(or)1177 4756 y Ff(Z)1260 4782 y Fl(1)1223 4945 y Fe(0)1351 4873 y Fo(e)1396 4832 y Fl(\000)p Fe(2)p Fk(\031)r(iy)1595 4873 y Fo(y)1647 4832 y Fk(z)s Fl(\000)p Fe(1)1776 4873 y Fo(dy)58 b Fq(=)d(\(2)p Fo(\031)t Fq(\))2248 4832 y Fl(\000)p Fk(z)2343 4873 y Fo(e)2388 4832 y Fl(\000)2453 4805 y Fg(\031)r(z)r(i)p 2453 4817 92 4 v 2483 4858 a Fj(2)2559 4873 y Fq(\000\()p Fo(z)t Fq(\))p Fo(:)545 b Fq(\(3.57\))1898 5214 y(22)p eop %%Page: 23 23 23 22 bop 328 631 a Fq(The)32 b(comparison)f(of)g(the)h(real)f(and)g (imaginary)e(part)j(of)f(the)h(b)s(oth)f(sides)h(of)f(\(.\))43 b(giv)m(es)714 832 y(~)687 857 y Fo(K)8 b Fq(\()p Fo(s)p Fq(\))55 b(=)g Fo(int)1212 816 y Fl(1)1212 882 y Fe(0)1287 857 y Fo(x)1342 816 y Fk(s)p Fl(\000)p Fe(1)1470 857 y Fo(cos)p Fq(\(2)p Fo(\031)t(x)p Fq(\))p Fo(dx)g Fq(=)g(\(2)p Fo(\031)t Fq(\))2320 816 y Fl(\000)p Fk(s)2412 857 y Fo(cos)p Fq(\()2595 790 y Fo(\031)t(s)p 2595 834 105 4 v 2622 926 a Fq(2)2709 857 y(\)\000\()p Fo(s)p Fq(\))p Fo(;)360 b Fq(\(3.58\))328 1086 y(and)1064 1089 y Ff(Z)1147 1116 y Fl(1)1110 1278 y Fe(0)1238 1206 y Fo(x)1293 1165 y Fk(s)p Fl(\000)p Fe(1)1421 1206 y Fo(sin)p Fq(\(2)p Fo(\031)t(x)p Fq(\))p Fo(dx)55 b Fq(=)h(\(2)p Fo(\031)t Fq(\))2274 1165 y Fl(\000)p Fk(s)2365 1206 y Fo(sin)p Fq(\()2550 1139 y Fo(\031)t(s)p 2550 1183 V 2578 1275 a Fq(2)2665 1206 y(\)\000\()p Fo(s)p Fq(\))p Fo(:)404 b Fq(\(3.59\))328 1408 y(F)-8 b(rom)31 b(\(.\))43 b(and)33 b(\(.\))43 b(w)m(e)34 b(obtain)d(that)1377 1609 y(~)1351 1634 y Fo(K)7 b Fq(\()p Fo(s)p Fq(\))83 b(=)g(\(2)p Fo(\031)t Fq(\))1989 1593 y Fl(\000)p Fk(s)2080 1634 y Fq(\000\()p Fo(s)p Fq(\))p Fo(cos)p Fq(\()2446 1567 y Fo(\031)t(s)p 2446 1611 V 2474 1702 a Fq(2)2561 1634 y(\))p Fo(:)691 b Fq(\(3.60\))328 1862 y(Moreo)m(v)m(er:)612 2040 y(~)586 2065 y Fo(K)7 b Fq(\()p Fo(s)p Fq(\))824 2040 y(~)798 2065 y Fo(K)g Fq(\(1)22 b Fn(\000)g Fo(s)p Fq(\))28 b(=)f(\(2)p Fo(\031)t Fq(\))1495 2024 y Fl(\000)p Fk(s)1587 2065 y Fq(\000\()p Fo(s)p Fq(\))p Fo(cos)1915 1998 y(\031)t(s)p 1915 2042 V 1942 2133 a Fq(2)2029 2065 y(\(2)p Fo(\031)t Fq(\))2213 2024 y Fk(s)p Fl(\000)p Fe(1)2340 2065 y Fq(\000\(1)21 b Fn(\000)i Fo(s)p Fq(\))p Fo(sin)2840 1998 y(\031)t(s)p 2840 2042 V 2868 2133 a Fq(2)2983 2065 y(=)258 b(\(3.61\))1164 2356 y(=)1307 2289 y(1)p 1278 2333 108 4 v 1278 2424 a(4)p Fo(\031)1396 2356 y Fq(2)p Fo(sin)1592 2289 y(\031)t(s)p 1592 2333 105 4 v 1620 2424 a Fq(2)1707 2356 y Fo(cos)1852 2289 y(\031)t(s)p 1852 2333 V 1879 2424 a Fq(2)1966 2356 y(\000\()p Fo(s)p Fq(\)\000\(1)22 b Fn(\000)g Fo(s)p Fq(\))28 b(=)2643 2289 y(1)p 2643 2333 49 4 v 2643 2424 a(4)2702 2356 y Fo(;)328 2554 y Fq(since)f(\000\()p Fo(s)p Fq(\)\000\(1)11 b Fn(\000)g Fo(s)p Fq(\))27 b(=)h Fo(\031)t(=sin\031)t (s)f Fq(\(according)f(to)h(the)g(w)m(ell-kno)m(wn)h Fr(Euler)i(form)m (ula)p Fq(\).)328 2675 y(Therefore,)k(follo)m(wing)c(\(.\))43 b(,)33 b(if)e Fn(j)c Fo(\025)h Fn(j6)p Fq(=)f(1)32 b(w)m(e)i(ha)m(v)m (e)700 2938 y(\()p Fo(M)10 b(f)h Fq(\)\()p Fo(s)p Fq(\))56 b(=)1258 2867 y(\()p Fo(M)10 b(g)1447 2882 y Fk(A)1504 2867 y Fq(\)\()p Fo(s)p Fq(\))22 b Fn(\000)h Fq(2)p Fo(\025)p Fq(\()p Fo(M)10 b(g)2081 2882 y Fk(A)2138 2867 y Fq(\)\(1)22 b Fn(\000)g Fo(s)p Fq(\)\(2)p Fo(\031)t Fq(\))2652 2831 y Fl(\000)p Fk(s)2743 2867 y Fo(cos)2888 2828 y Fk(\031)r(s)p 2888 2844 76 4 v 2908 2901 a Fe(2)2974 2867 y Fq(\000\()p Fo(s)p Fq(\))p 1258 2915 1899 4 v 2074 3006 a(1)g Fn(\000)g Fo(\025)2301 2977 y Fe(2)3166 2938 y Fo(:)328 3174 y Fq(Hence)429 3400 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))28 b(=)961 3332 y(1)p 760 3377 451 4 v 760 3468 a(2)p Fo(\031)t Fq(\(1)22 b Fn(\000)h Fo(\025)1134 3439 y Fe(2)1173 3468 y Fq(\))1237 3283 y Ff(Z)1320 3309 y Fk(\033)r Fe(+)p Fk(i)p Fl(1)1283 3471 y Fk(\033)r Fl(\000)p Fk(i)p Fl(1)1517 3400 y Fq(\()p Fo(M)10 b(g)1706 3415 y Fk(A)1763 3400 y Fq(\()p Fo(s)p Fq(\))22 b Fn(\000)h Fq(2)p Fo(\025)p Fq(\(2)p Fo(\031)t Fq(\))2297 3359 y Fl(\000)p Fk(s)2388 3400 y Fo(cos)2533 3332 y(\031)t(s)p 2533 3377 105 4 v 2561 3468 a Fq(2)2647 3400 y(\000\()p Fo(s)p Fq(\)\))p Fo(x)2923 3359 y Fl(\000)p Fk(s)3015 3400 y Fo(ds)k Fq(=)102 b(\(3.62\))328 3721 y(=)550 3653 y(1)p 441 3698 267 4 v 441 3789 a(1)22 b Fn(\000)h Fo(\025)669 3760 y Fe(2)718 3721 y Fn(\001)802 3653 y Fq(1)p 756 3698 141 4 v 756 3789 a(2)p Fo(\031)t(i)923 3604 y Ff(Z)1006 3630 y Fk(\033)r Fe(+)p Fk(i)p Fl(1)970 3792 y Fk(\033)r Fl(\000)p Fk(i)p Fl(1)1203 3721 y Fq(\()p Fo(M)10 b(g)1392 3736 y Fk(A)1449 3721 y Fq(\)\()p Fo(s)p Fq(\))p Fo(x)1664 3680 y Fl(\000)p Fk(s)1756 3721 y Fo(ds)p Fq(+)2019 3653 y(2)p Fo(\025)p 1939 3698 267 4 v 1939 3789 a Fq(1)22 b Fn(\000)g Fo(\025)2166 3760 y Fe(2)2215 3721 y Fn(\001)2299 3653 y Fq(1)p 2253 3698 141 4 v 2253 3789 a(2)p Fo(\031)t(i)2421 3604 y Ff(Z)2504 3630 y Fk(\033)r Fe(+)p Fk(i)p Fl(1)2467 3792 y Fk(\033)r Fl(\000)p Fk(i)p Fl(1)2700 3721 y Fq(\()p Fo(M)10 b(g)2889 3736 y Fk(A)2946 3721 y Fq(\(1)p Fn(\000)p Fo(s)p Fq(\)\(2)p Fo(\031)t Fq(\))3378 3680 y Fl(\000)p Fk(s)3470 3721 y Fo(cos)3615 3653 y(\031)t(s)p 3615 3698 105 4 v 3643 3789 a Fq(2)3729 3721 y(\000\()p Fo(s)p Fq(\))p Fo(x)3967 3680 y Fl(\000)p Fk(s)4059 3721 y Fo(ds)p Fq(\))p Fo(:)328 3929 y Fq(In)34 b(the)f(righ)m(t-hand)g(side)g(of)g (\(.\),)g(let's)g(replace)h(in)e(the)i(second)g(in)m(tegral)e Fo(M)10 b(g)3215 3944 y Fk(A)3272 3929 y Fq(\(1)23 b Fn(\000)g Fo(s)p Fq(\))328 4049 y(through)696 3978 y Ff(R)752 4005 y Fl(1)736 4075 y Fe(0)843 4049 y Fo(g)890 4064 y Fk(A)947 4049 y Fq(\()p Fo(x)p Fq(\))p Fo(x)1133 4013 y Fl(\000)p Fk(s)1225 4049 y Fo(dx)33 b Fq(and)g(observ)m(e)h (that)1267 4233 y(1)p 1221 4278 141 4 v 1221 4369 a(2)p Fo(\031)t(i)1388 4184 y Ff(Z)1471 4210 y Fk(\033)r Fe(+)p Fk(i)p Fl(1)1434 4372 y Fk(\033)r Fl(\000)p Fk(i)p Fl(1)1667 4301 y Fq(\()p Fo(M)10 b(g)1856 4316 y Fk(A)1914 4301 y Fq(\)\()p Fo(s)p Fq(\))p Fo(x)2129 4260 y Fl(\000)p Fk(s)2221 4301 y Fo(ds)55 b Fq(=)g Fo(g)2551 4316 y Fk(A)2608 4301 y Fq(\()p Fo(x)p Fq(\))p Fo(:)328 4545 y Fq(The)33 b(form)m(ula)e(\(.\))43 b(obtain)32 b(no)m(w)h(the)g(follo)m(wing)d (form:)426 4803 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))28 b(=)773 4736 y Fo(g)820 4751 y Fk(A)877 4736 y Fq(\()p Fo(x)p Fq(\))p 757 4780 267 4 v 757 4872 a(1)22 b Fn(\000)h Fo(\025)985 4843 y Fe(2)1056 4803 y Fq(+)1245 4736 y(2)p Fo(\025)p 1164 4780 V 1164 4872 a Fq(1)f Fn(\000)h Fo(\025)1392 4843 y Fe(2)1463 4803 y Fn(\001)1569 4736 y Fq(1)p 1523 4780 141 4 v 1523 4872 a(2)p Fo(\031)t(i)1691 4686 y Ff(Z)1774 4712 y Fk(\033)r Fe(+)p Fk(i)p Fl(1)1737 4875 y Fk(\033)r Fl(\000)p Fk(i)p Fl(1)1970 4803 y Fq(\(2)p Fo(\031)t Fq(\))2154 4762 y Fl(\000)p Fk(s)2245 4803 y Fo(cos)2390 4736 y(\031)t(s)p 2390 4780 105 4 v 2418 4872 a Fq(2)2505 4803 y(\000\()p Fo(s)p Fq(\)\()p Fo(xy)t Fq(\))2871 4762 y Fl(\000)p Fk(s)2978 4686 y Ff(Z)3061 4712 y Fl(1)3025 4875 y Fe(0)3153 4803 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)t(:)3317 4965 y Fq(\(3.63\))1898 5214 y(23)p eop %%Page: 24 24 24 23 bop 328 631 a Fq(According)32 b(to)g(the)h(in)m(v)m(ersion)g (form)m(ula)e(for)h(the)h(Mellin)e(transform)g(w)m(e)j(get)953 826 y(1)p 906 870 141 4 v 906 961 a(2)p Fo(\031)t(i)1074 776 y Ff(Z)1157 802 y Fk(\033)r Fe(+)p Fk(i)p Fl(1)1120 965 y Fk(\033)r Fl(\000)p Fk(i)p Fl(1)1353 893 y Fq(\(2)p Fo(\031)t Fq(\))1537 852 y Fl(\000)p Fk(s)1629 893 y Fq(\000\()p Fo(s)p Fq(\))p Fo(cos)1957 826 y(\031)t(s)p 1957 870 105 4 v 1984 961 a Fq(2)2071 893 y(\()p Fo(xy)t Fq(\))2254 852 y Fl(\000)p Fk(s)2345 893 y Fo(ds)55 b Fq(=)g Fo(cos)p Fq(\(2)p Fo(\031)t(xy)t Fq(\))p Fo(;)328 1149 y Fq(.)43 b(Therefore)34 b(,)f(\014nally)e(w)m(e)j(obtain)910 1418 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))83 b(=)1368 1351 y Fo(g)1415 1366 y Fk(A)1472 1351 y Fq(\()p Fo(x)p Fq(\))p 1352 1395 267 4 v 1352 1486 a(1)22 b Fn(\000)h Fo(\025)1580 1458 y Fe(2)1651 1418 y Fq(+)1840 1351 y(2)p Fo(\025)p 1759 1395 V 1759 1486 a Fq(1)f Fn(\000)g Fo(\025)1986 1458 y Fe(2)2052 1301 y Ff(Z)2135 1327 y Fl(1)2098 1489 y Fe(0)2227 1418 y Fo(g)2274 1433 y Fk(A)2330 1418 y Fq(\()p Fo(y)t Fq(\))p Fo(cos)p Fq(\(2)p Fo(\031)t(xy)t Fq(\))p Fo(dy)t(:)303 b Fq(\(3.64\))328 1761 y Fp(4)161 b(F)-13 b(rom)35 b(\(RH\)-quasi-\014xed)h(to)f(\(RH\)-\014xed)g(p)t (oin)l(ts)570 1943 y(of)54 b Fa(F)328 2162 y Fq(According)31 b(to)g(the)h(Prop)s(osition)e(3.1,)i(for)f(eac)m(h)h(CID)g(,)f Fr(bimo)s(dal)f Fq(amplitude)g Fo(A)i Fq(and)328 2283 y(real)25 b Fo(\025)h Fq(with)g Fn(j)i Fo(\025)f Fn(j6)p Fq(=)g(1,)h(there)f(exists)g(a)f(\(RH\))g(-quasi-\014xed)g(p)s(oin)m(t) g Fo(F)2839 2298 y Fk(\025)2912 2283 y Fq(=)h Fo(f)3063 2298 y Fk(\025)3118 2283 y Fq(+)9 b Fo(\025)3281 2256 y Fq(^)3260 2283 y Fo(f)3308 2298 y Fk(\025)3380 2283 y Fq(of)25 b Fn(F)328 2403 y Fq(with)32 b(a)g Fr(symmetric)f Fo(f)1224 2418 y Fk(\025)1301 2403 y Fq(from)h Fn(S)7 b Fq(\()p Fm(R)i Fq(\),)33 b(and)f(suc)m(h)i(that)f(it)e(solv)m(es)j (the)f(F)-8 b(o)m(x)32 b(equation:)1448 2623 y Fo(F)1511 2638 y Fk(\025)1557 2623 y Fq(\()p Fo(x)p Fq(\))c(=)f Fo(f)1867 2638 y Fk(\025)1913 2623 y Fq(\()p Fo(x)p Fq(\))22 b(+)g Fo(\025)2242 2597 y Fq(^)2221 2623 y Fo(f)2269 2638 y Fk(\025)2314 2623 y Fq(\()p Fo(x)p Fq(\))872 b(\(4.65\))1204 2843 y(=)28 b Fo(G)p Fq(\()p Fo(x)p Fq(\))22 b(+)g Fo(A)p Fq(\()p Fo(x)p Fq(\))56 b(=:)f Fo(g)2101 2858 y Fk(A)2158 2843 y Fq(\()p Fo(x)p Fq(\))p Fo(;)45 b(x)28 b Fn(2)g Fm(R)2603 2858 y Fe(+)2662 2843 y Fo(:)328 3017 y Fq(and)1650 3138 y Fo(F)1713 3153 y Fk(\025)1759 3138 y Fq(\(0\))55 b(=)g Fo(G)p Fq(\(0\))p Fo(:)1018 b Fq(\(4.66\))474 3312 y(Th)m(us,)34 b(there)g(exists)f(a)f Fr(whole)37 b(family)1379 3532 y Fo(F)14 b Fq(\()p Fo(x)p Fq(\))55 b(:=)h Fn(f)p Fo(F)1914 3547 y Fk(\025)1959 3532 y Fq(\()p Fo(x)p Fq(\))g(:)27 b Fn(j)h Fo(\025)f Fn(j6)p Fq(=)h(1)p Fn(g)328 3752 y Fq(,)42 b(of)e(quasi-\014xed)i(p)s(oin)m(ts)e Fo(F)1377 3767 y Fk(\025)1463 3752 y Fq(=)h Fo(f)1628 3767 y Fk(\025)1701 3752 y Fq(+)28 b Fo(\025)1883 3726 y Fq(^)1862 3752 y Fo(f)1910 3767 y Fk(\025)1955 3752 y Fq(,)42 b(whic)m(h)f(constitutes)h (a)e(material)d(\(stu\013)7 b(\))328 3873 y(for)33 b(obtaining)e(a)i (\()p Fo(R)q(H)8 b Fq(\)-\014xed)34 b(p)s(oin)m(t)e(of)h Fn(F)43 b Fq(-)33 b(let's)g(write)g(-)g Fo(F)2637 3888 y Fe(1)2710 3873 y Fq(-)g(the)g(main)f(tec)m(hnical)328 3993 y(ob)5 b(ject)45 b(of)g(this)f(pap)s(er.)80 b(Let's)45 b(observ)m(e)i(also)c(that)i(is)f(v)m(ery)i(imp)s(ortan)m(t)d(the)i (exact)328 4113 y(structure)34 b(of)e Fo(f)905 4128 y Fk(\025)983 4113 y Fq(\(cf.\(3.64\))1185 4388 y Fo(f)1233 4403 y Fk(\025)1278 4388 y Fq(\()p Fo(x)p Fq(\))56 b(=)1622 4321 y Fo(g)1669 4336 y Fk(A)1725 4321 y Fq(\()p Fo(x)p Fq(\))p 1606 4365 V 1606 4457 a(1)22 b Fn(\000)g Fo(\025)1833 4428 y Fe(2)1905 4388 y Fq(+)2118 4321 y Fo(\025)p 2013 4365 V 2013 4457 a Fq(1)g Fn(\000)g Fo(\025)2240 4428 y Fe(2)2290 4388 y Fq(\()p Fn(F)10 b Fo(g)2457 4403 y Fk(A)2513 4388 y Fq(\)\()p Fo(x)p Fq(\))p Fo(:)608 b Fq(\(4.67\))328 4648 y(F)-8 b(rom)31 b(\(4.67\))h(w)m(e)h(immediately)d (get)1056 4868 y Fo(f)1104 4883 y Fk(\025)1149 4868 y Fq(\()p Fo(x)p Fq(\))22 b(+)g Fo(f)1448 4883 y Fe(1)p Fk(=\025)1564 4868 y Fq(\()p Fo(x)p Fq(\))56 b(=)f Fo(g)1929 4883 y Fk(a)1971 4868 y Fq(\()p Fo(x)p Fq(\))28 b(;)44 b Fo(x)28 b Fn(2)g Fm(R)2444 4883 y Fe(+)2503 4868 y Fo(;)17 b Fn(j)27 b Fo(\025)g Fn(j6)p Fq(=)h(1)p Fo(:)423 b Fq(\(4.68\))1898 5214 y(24)p eop %%Page: 25 25 25 24 bop 328 631 a Fq(In)23 b(the)h(family)d(of)i(the)g(equations)h (\(4.67\))e(are)h(also)g(written)g(the)g(imp)s(ortan)m(t)f Fr(algebraic)328 751 y Fq(expression)36 b(on)g Fo(\025)f Fq(.)51 b(Let)36 b Fo(I)1330 780 y Fm(R)1400 751 y Fq(\()p Fo(\025)p Fq(\))c(=)g(1)p Fo(=\025)i Fq(b)s(e)i(the)f Fr(in)m(v)m(erse)g Fq(\(hip)s(erb)s(olic\))e(map)h(on)i Fm(R)3499 715 y Fl(\003)3539 751 y Fq(.)328 872 y(Observ)m(e)e(that)f (the)g(function)1568 1110 y Fn(I)1622 1138 y Fm(R)1692 1110 y Fq(\()p Fo(\025)p Fq(\))55 b(=)2182 1043 y Fo(\025)p 2021 1087 378 4 v 2021 1178 a Fn(j)28 b Fq(1)22 b Fn(\000)g Fo(\025)2304 1150 y Fe(2)2371 1178 y Fn(j)3317 1110 y Fq(\(4.69\))328 1380 y(is)f Fo(I)458 1408 y Fm(R)528 1380 y Fr(-in)m(v)-6 b(arian)m(t)21 b Fq(on)h Fm(R)1210 1343 y Fl(\003)1210 1404 y Fe(+)1270 1380 y Fn(\000)q(f)p Fq(1)p Fn(g)p Fq(,)h(i.e.)40 b Fn(I)1766 1408 y Fm(R)1836 1380 y Fq(\()p Fo(I)1917 1408 y Fm(R)1986 1380 y Fq(\()p Fo(\025)p Fq(\)\))28 b(=)f Fn(I)2342 1408 y Fm(R)2412 1380 y Fq(\()p Fo(\025)p Fq(\).)40 b(Moreo)m(v)m(er,)26 b(the)c(function)328 1500 y Fn(j)33 b(I)443 1528 y Fm(R)513 1500 y Fq(\()p Fo(\025)p Fq(\))f Fn(j)h Fo(=\025)i Fq(is)h(the)g Fr(densit)m(y)f Fq(of)g(a)h Fr(\014nite)k(Borel)g(measure)c Fo(\026)2831 1515 y Fk(I)2906 1500 y Fq(on)g Fm(R)3110 1464 y Fl(\003)3110 1525 y Fe(+)3205 1500 y Fq(giv)m(en)g(b)m(y)328 1621 y(the)d(form)m(ula)1376 1765 y Fo(\026)1435 1780 y Fk(I)1475 1765 y Fq(\()p Fo(B)5 b Fq(\))55 b(:=)g Fo(V)5 b(:p:)2025 1648 y Ff(Z)2071 1837 y Fk(B)2294 1698 y Fo(d\025)p 2159 1742 V 2159 1834 a Fn(j)27 b Fq(1)22 b Fn(\000)g Fo(\025)2441 1805 y Fe(2)2508 1834 y Fn(j)2546 1765 y Fo(;)744 b Fq(\(4.70\))328 1987 y(where)32 b(the)g(in)m(tegral)d (exists)j(in)e(the)i(sense)g(of)f(the)g(Cauc)m(h)m(y's)j Fr("V)-9 b(aleur)34 b(principale")328 2108 y Fq(in)e(1)g (\(cf.e.g.[Fi,XI)s(I)s(I.484]\))g(,)g(and)h Fo(B)38 b Fq(is)32 b(a)g(Borel)g(set)h(in)f Fm(R)2538 2072 y Fl(\003)2538 2132 y Fe(+)2597 2108 y Fq(.)474 2228 y(The)23 b(measure)f Fo(\026)1092 2243 y Fk(I)1153 2228 y Fq(is)g(the)g Fr(in)m(v)-6 b(arian)m(t)23 b(measure)g Fq(of)e(the)h Fr(top)s(ologically-dynamical) 328 2349 y(system)32 b Fq(\()p Fm(R)800 2312 y Fl(\003)800 2373 y Fe(+)859 2349 y Fo(;)17 b Fn(f)p Fo(I)1004 2312 y Fk(n)996 2388 y Fm(R)1093 2349 y Fq(:)28 b Fo(n)f Fn(2)i Fm(N)12 b Fn(g)p Fq(\))32 b(,)h(i.e.)1476 2574 y Fo(\026)1535 2589 y Fk(I)1575 2574 y Fq(\(\()p Fo(I)1702 2533 y Fk(n)1694 2609 y Fm(R)1763 2574 y Fq(\))1801 2533 y Fl(\000)p Fe(1)1896 2574 y Fq(\()p Fo(B)5 b Fq(\)\))55 b(=)g Fo(\026)2334 2589 y Fk(I)2374 2574 y Fq(\()p Fo(B)5 b Fq(\))p Fo(;)761 b Fq(\(4.71\))328 2788 y(for)32 b(eac)m(h)h Fo(n)28 b Fn(2)g Fm(N)46 b Fq(and)32 b(a)h(Borel)e(set)j(B.)474 2908 y(According)i(to)g(the)h(ab)s(o)m(v)m(e)g(notations)f(,)h(the)g (comp)s(onen)m(t)g Fo(f)2736 2923 y Fk(\025)2817 2908 y Fq(of)f(the)h(\014xed)h(p)s(oin)m(t)328 3029 y Fo(F)391 3044 y Fk(\025)469 3029 y Fq(w)m(e)c(can)e(write)h(in)e(the)i(form:) 1167 3287 y Fo(f)1215 3302 y Fk(\025)1260 3287 y Fq(\()p Fo(x)p Fq(\))56 b(=)1588 3220 y Fo(d\026)1698 3235 y Fk(I)1738 3220 y Fq(\()p Fo(\025)p Fq(\))p 1588 3264 283 4 v 1675 3356 a Fo(d\025)1880 3287 y(g)1927 3302 y Fk(A)1984 3287 y Fq(\()p Fo(x)p Fq(\))22 b(+)g Fn(I)2289 3316 y Fm(R)2360 3287 y Fq(\()p Fo(\025)p Fq(\))27 b(^)-76 b Fo(g)2540 3302 y Fk(A)2596 3287 y Fq(\()p Fo(x)p Fq(\))p Fo(:)563 b Fq(\(4.72\))328 3540 y Fr(Remark)37 b(4)49 b Fb(A)-5 b(c)g(c)g(or)g(ding)24 b(to)h(the)g(classic)-5 b(al)24 b Fr(Bogolub)s(o\013-Krilo\013)f(theorem)h Fb(\(cf.[BK]\))328 3660 y(every)37 b(top)-5 b(olo)g(gic)g(al,)37 b(dynamic)-5 b(al)37 b(system)g Fn(f)p Fo(T)1998 3624 y Fk(n)2044 3660 y Fn(g)h Fb(gener)-5 b(ate)g(d)36 b(by)i(a)f(map)g Fo(T)46 b Fq(:)33 b Fo(X)40 b Fn(\000)-16 b(!)32 b Fo(X)328 3780 y Fb(de\014ne)-5 b(d)38 b(on)g(a)h(c)-5 b(omp)g(act)38 b(metric)g(sp)-5 b(ac)g(e)38 b Fo(X)8 b Fb(,)40 b(admits)e(an)h Fr(in)m(v)-6 b(arian)m(t)41 b(measure)81 b Fo(\026)3481 3795 y Fk(T)3536 3780 y Fb(.)328 3901 y(In)35 b(the)g(se)-5 b(quel)35 b(the)h(me)-5 b(asur)g(e)35 b Fo(\026)1500 3916 y Fk(T)1590 3901 y Fb(is)g(c)-5 b(al)5 b(le)-5 b(d)35 b(the)g Fr(Bogolub)s(o\013-Krilo\013)i(measure)f Fb(of)328 4021 y(a)f(\(top)-5 b(olo)g(gic)g(al)5 b(ly\))33 b(dynamic)-5 b(al)34 b(system)h Fq(\()p Fo(X)r(;)17 b(T)d Fq(\))p Fb(.)474 4242 y Fq(The)46 b(constructed)g(ab)s(o)m(v)m(e)g(system)g Fn(f)p Fo(F)1963 4257 y Fk(\025)2008 4242 y Fn(g)f Fq(of)f (\(RH\)-quasi-\014xed)h(p)s(oin)m(ts)f(,)k(com-)328 4363 y(monly)c(with)i(the)g Fr(Bogolub)s(o\013-Krilo\013)k(measure)c Fo(\026)2494 4378 y Fk(H)2607 4363 y Fq(of)f Fo(I)2774 4378 y Fk(H)2841 4363 y Fq(\()p Fo(\025)p Fq(\))50 b(=)3162 4324 y Fe(1)p 3159 4340 42 4 v 3159 4397 a Fk(\025)3211 4363 y Fo(;)17 b(\025)49 b Fn(2)h Fo(H)328 4483 y Fq(\(where)34 b Fo(H)41 b Fq(is)32 b(the)i(\014eld)f(of)g Fr(quaternions)p Fq(\),p)s(ermits)f(us)i(to)e(pro)m(v)m(e)j(the)e(main)f(tec)m(hni-)328 4604 y(cal)g(result)g(of)g(this)h(pap)s(er)f(:)44 b(the)33 b(existence)h(of)e(\(RH\)-\014xed)h(p)s(oin)m(ts)f(of)g Fn(F)10 b Fq(.)474 4724 y(W)-8 b(e)42 b(start)f(from)f(some)h (preliminary)e(results)j(concerning)f Fr(in)m(tegration)f Fq(in)g(the)328 4844 y(area)33 b(of)g Fr(lo)s(cal)38 b(\014elds)p Fq(.)46 b(\(Let)34 b(recall)e(that)h(our)h(Brou)m(w)m(er)h (logic)c(pro)s(of)i(of)g(\(RH\))h(giv)m(en)328 4965 y(in)39 b([M)s(\030)-46 b(a2])39 b(is)g(strongly)g(based)h(-)f(as)h(it)e(w)m (as)i(observ)m(ed)i(b)m(y)e(W.)g(Narkiewicz)f(-)g(on)g(the)1898 5214 y(25)p eop %%Page: 26 26 26 25 bop 328 631 a Fq(in)m(tegration)39 b(in)i(the)h (in\014nite-dimensional)37 b(function)k(spaces\).)71 b(The)42 b(idea)e(of)h(using)328 751 y(lo)s(cal)31 b(\014elds)j(in)f (the)h(con)m(text)h(of)f(the)g(Riemann)e(h)m(yp)s(othesis)j(is)e(not)h (new.)48 b(In)34 b(fact)f Fr(de)328 872 y(Branges)27 b Fq(in)e(the)h(men)m(tioned)g(in)g(the)g(in)m(tro)s(duction)f(-)g (series)i(of)f(pap)s(ers,)i(sho)m(w)m(ed)g(that)328 992 y Fo(p)p Fr(-adic)37 b(analysis)c Fq(is)f(strictly)g(connected)i(with)e (\()p Fo(R)q(H)8 b Fq(\).)474 1112 y(The)42 b(main)d(idea)h(dev)m(elop) s(es)i(here)f(b)s(elo)m(w,)h(is)f(based)g(on)g(the)g(observ)-5 b(ation,)42 b(that)328 1233 y(suitable)34 b(results)h(concerning)g(the) h Fr(transition)d Fq(from)h(the)h(family)d Fn(f)p Fo(f)2977 1248 y Fk(\025)3054 1233 y Fq(:)g Fo(\025)f(>)h Fq(0)p Fo(;)17 b(\025)31 b Fn(6)p Fq(=)328 1353 y(1)p Fn(g)j Fq(to)g Fo(f)630 1368 y Fe(1)670 1353 y Fq(,)h(whic)m(h)g(normally)e (needs)j(analysis)e(on)g Fm(R)40 b Fq(=:)31 b Fm(Q)2501 1373 y Fl(1)2576 1353 y Fq(,)k(can)g(b)s(e)g Fr(algebraically)328 1474 y Fq(obtained.)474 1594 y(Let)40 b Fo(K)47 b Fq(b)s(e)40 b(an)g(arbitrary)f Fr(lo)s(cal)44 b(\014eld)c Fq(,)h(i.e.)65 b(in)39 b(the)h(W)-8 b(eil's)39 b Fr(Basic)45 b(Num)m(b)s(er)328 1714 y(Theory)28 b Fq(terminology)-8 b(,)27 b(it)h(is)f(an)i(arbitrary) e(non-discrete)h(Lo)s(cally)f(Compact)h(\(LC)h(in)328 1835 y(short\))k(\014eld)f(\(not)g(necessary)j Fr(comm)m(utativ)m(e)p Fq(\)\(cf.e.g.)42 b([W)-8 b(e]\).)474 1955 y(Let)40 b Fn(j)g Fo(:)f Fn(j)818 1970 y Fk(K)926 1955 y Fq(denotes)i(the)f (canonica)f(non-trivial)e Fr(v)-6 b(alution)39 b Fq(\(absolute)g(v)-5 b(alue)39 b(or)328 2076 y(p)s(oin)m(t)34 b(of)h Fo(K)7 b Fq(\),)36 b(whic)m(h)g(can)f(b)s(e)h Fr(arc)m(himedean)f Fq(\(in)f(the)i(case)g(of)f Fr(connected)h Fq(\014elds\))328 2196 y(or)c Fr(non-arc)m(himedean)i Fq(\(in)d(the)i(case)h(of)e Fr(totally-disconnected)f Fq(\014elds\).)474 2316 y(Let)24 b Fo(H)721 2331 y Fk(K)813 2316 y Fq(b)s(e)g(the)g Fr(Haar)j(measure)e Fq(on)e(\()p Fo(K)r(;)17 b Fq(+\))23 b(\()g(the)i Fr(left)h(Haar)h (measure)d Fq(in)f(the)328 2437 y(case)j(when)g Fo(K)32 b Fq(is)24 b(non-comm)m(utativ)m(e\).)40 b(Let's)25 b(denote)h(:)39 b Fo(R)2504 2452 y Fk(K)2601 2437 y Fq(:=)27 b Fn(f)p Fo(k)k Fn(2)d Fo(K)35 b Fq(:)p Fn(j)27 b Fo(k)k Fn(j)3267 2452 y Fk(K)3335 2437 y Fn(\024)d Fq(1)p Fn(g)p Fq(.)328 2557 y(It)33 b(is)f(v)m(ery)i(con)m(v)m(enien)m(t)g(to)e(normalize)f Fo(H)1873 2572 y Fk(K)1973 2557 y Fq(in)h(the)h(follo)m(wing)d (standard)j(w)m(a)m(y)h(:)328 2677 y(\(i\))g(if)g Fo(K)43 b Fq(is)35 b(non-arc)m(himedean)f(then)i Fo(R)1842 2692 y Fk(K)1946 2677 y Fq(is)f(the)h(maximal)c(compact)j(subring)g(of)g Fo(K)328 2798 y Fq(\(the)e(ring)e(of)h(in)m(tegers)h(of)g(K\).)f(W)-8 b(e)33 b(assume)g(that)f Fo(H)2288 2813 y Fk(K)2357 2798 y Fq(\()p Fo(R)2469 2813 y Fk(K)2537 2798 y Fq(\))c(=)f(1.)328 2918 y(\(ii\))32 b(in)i(the)h(arc)m(himedean)f(case)h(w)m(e)g(ob)m (viously)f(ha)m(v)m(e:)49 b Fo(H)2480 2947 y Fm(R)2549 2918 y Fq(\()p Fo(R)2661 2947 y Fm(R)2731 2918 y Fq(\))31 b(=)g(2)p Fo(;)17 b(H)3081 2947 y Fm(C)3150 2918 y Fq(\()p Fo(R)3262 2947 y Fm(C)3332 2918 y Fq(\))30 b(=)h Fo(\031)328 3039 y Fq(and)i Fo(H)599 3054 y Fk(H)666 3039 y Fq(\()p Fo(R)t Fq(\))28 b(=)f(7)p Fo(\031)t(=)p Fq(4.)474 3159 y(Eac)m(h)48 b Fr(automorphism)d Fo(\025)i Fq(of)f(\()p Fo(K)r(;)17 b Fq(+\))45 b(c)m(hange)j(the)f(Haar)f(measure)h Fo(H)3286 3174 y Fk(K)3401 3159 y Fq(in)m(to)328 3279 y Fo(cH)451 3294 y Fk(K)554 3279 y Fq(with)34 b Fo(c)d Fn(2)h Fm(R)1014 3243 y Fl(\003)1014 3304 y Fe(+)1108 3279 y Fq(\(since)j(ob)m(viously)g(according)f(to)g(the)h(Neumann-W)-8 b(eil)33 b(theorem)328 3400 y(a)e(Haar)f(measure)h(is)g(unique)g(up)g (to)g(the)g(constan)m(t\).)44 b(The)32 b(n)m(um)m(b)s(er)f Fo(c)g Fq(do)g(not)g(dep)s(end)328 3520 y(on)f(the)i(c)m(hoice)f(of)f (a)g(Haar)g(measure)h(and)g(w)m(e)h(denote)f(it)f(b)m(y)h(\001)2673 3535 y Fk(K)2742 3520 y Fq(\()p Fo(\025)p Fq(\))f(or)h Fo(mod)3206 3535 y Fk(K)3274 3520 y Fq(\()p Fo(\025)p Fq(\).)42 b(In)328 3641 y(the)36 b(other)f(w)m(ords,)i(it)e(is)f (de\014ned)j(b)m(y)f(the)g(one)g(of)e(the)i(b)s(elo)m(w)f(\(equiv)-5 b(alen)m(t\))35 b(form)m(ulas)328 3761 y(\(cf.[W)-8 b(e,I.2]\))33 b(:)1384 3881 y Fo(H)1465 3896 y Fk(K)1533 3881 y Fq(\()p Fo(\025)p Fq(\()p Fo(B)5 b Fq(\)\))55 b(=)g(\001)2088 3896 y Fk(K)2157 3881 y Fq(\()p Fo(\025)p Fq(\))p Fo(H)2371 3896 y Fk(K)2439 3881 y Fq(\()p Fo(B)5 b Fq(\))p Fo(;)328 4056 y Fq(or)984 4059 y Ff(Z)1083 4176 y Fo(f)11 b Fq(\()p Fo(\025)1237 4135 y Fl(\000)p Fe(1)1331 4176 y Fq(\()p Fo(x)p Fq(\)\))p Fo(dH)1632 4191 y Fk(K)1700 4176 y Fq(\()p Fo(x)p Fq(\))56 b(=)f(\001)2099 4191 y Fk(K)2168 4176 y Fq(\()p Fo(\025)p Fq(\))2318 4059 y Ff(Z)2417 4176 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))p Fo(dH)2739 4191 y Fk(K)2807 4176 y Fq(\()p Fo(x)p Fq(\))p Fo(;)328 4377 y Fq(where)36 b Fo(B)41 b Fq(is)34 b(an)m(y)i(Borel)e(set)i(and)f Fo(f)46 b Fq(is)35 b(an)g(in)m(tegrable)f(function)h(with)3019 4306 y Ff(R)3091 4377 y Fo(f)11 b(dH)3282 4392 y Fk(K)3382 4377 y Fn(6)p Fq(=)32 b(0.)328 4497 y(The)40 b(second)g(form)m(ula)c(w) m(e)k(can)f(sym)m(b)s(olically)d(write)j(in)e(the)j(form)d(:)56 b Fo(dH)3120 4512 y Fk(K)3188 4497 y Fq(\()p Fo(\025)p Fq(\()p Fo(x)p Fq(\)\))38 b(=)328 4617 y(\001)409 4632 y Fk(K)478 4617 y Fq(\()p Fo(x)p Fq(\)\()p Fo(\025)p Fq(\))p Fo(dH)874 4632 y Fk(K)942 4617 y Fq(\()p Fo(x)p Fq(\).)474 4738 y(Let)49 b Fo(L)731 4753 y Fk(a)773 4738 y Fq(\()p Fo(x)p Fq(\))54 b(=)g Fo(a)33 b Fn(\001)f Fo(x)p Fq(;)17 b Fo(a)55 b Fn(6)p Fq(=)f(0)p Fo(;)17 b(x)54 b Fn(2)h Fo(K)g Fq(b)s(e)48 b(the)h(linear)d(m)m(ultiplication)e(\(b)m (y)49 b Fo(a)p Fq(\))1898 5214 y(26)p eop %%Page: 27 27 27 26 bop 328 631 a Fq(automorphism)30 b(of)i Fo(K)7 b Fq(.)44 b(W)-8 b(e)33 b(put)1321 834 y(\001)1402 849 y Fk(K)1470 834 y Fq(\()p Fo(a)p Fq(\))83 b(:=)g(\001)1947 849 y Fk(K)2016 834 y Fq(\()p Fo(L)2120 849 y Fk(a)2162 834 y Fq(\);)100 b Fo(a)27 b Fn(2)i Fo(K)2590 793 y Fl(\003)2629 834 y Fo(:)328 1038 y Fq(By)d Fo(I)517 1053 y Fk(K)586 1038 y Fq(\()p Fo(\025)p Fq(\))h(=)h Fo(\025)907 1002 y Fl(\000)p Fe(1)1001 1038 y Fq(;)17 b Fo(\025)27 b Fn(2)h Fo(K)1313 1002 y Fl(\003)1378 1038 y Fq(w)m(e)f(denote)g(the)f(the)g Fr(in)m(v)m(ersion)i(of)i Fo(K)2821 1002 y Fl(\003)2860 1038 y Fq(,)e(i.e.)40 b(the)26 b Fr(crucial)328 1158 y Fq(\(top)s(ologically-algebrai)o(c\))34 b(automorphism)j(of)i Fo(K)2257 1122 y Fl(\003)2336 1158 y Fq(of)g(order)g(2)g(:)h Fo(I)2921 1122 y Fe(2)2913 1183 y Fk(K)3059 1158 y Fq(=)79 b Fo(id)3298 1173 y Fk(K)3363 1154 y Fd(\003)3402 1158 y Fq(.)64 b(It)328 1288 y(is)43 b(w)m(ell-kno)m(wn)h(\(cf.[W)-8 b(e]\))44 b(that)1596 1241 y Fk(dH)1690 1252 y Fg(K)1749 1241 y Fe(\()p Fk(\025)p Fe(\))p 1596 1264 250 4 v 1614 1322 a(\001)1673 1333 y Fg(K)1732 1322 y Fe(\()p Fk(\025)p Fe(\))1899 1288 y Fq(is)f(a)g Fr(Haar)51 b(measure)44 b Fq(of)f(\()p Fo(K)3089 1252 y Fl(\003)3128 1288 y Fo(;)17 b Fn(\001)p Fq(\))43 b(\(cf.e.g.)328 1422 y([W)-8 b(e,VI)s(I.4,)33 b(Lemma5]\).)474 1542 y(Let's)k(also)d(men)m(tion)h(at)h(this)f(place,) h(that)g(in)e(the)j(most)e(imp)s(ortan)m(t)e(cases)38 b Fn(j)32 b(\001)h(j)3498 1557 y Fk(K)328 1662 y Fq(and)27 b(\001)593 1677 y Fk(K)688 1662 y Fq(coincides.)41 b(In)27 b(particular,)f(\001)1805 1691 y Fm(R)1903 1662 y Fq(=)p Fn(j)h Fo(:)h Fn(j)2117 1691 y Fm(R)2186 1662 y Fq(=)p Fn(j)f Fo(:)h Fn(j)2400 1677 y Fl(1)2501 1662 y Fq(and)e(for)g(all)f (p-adic)g(n)m(um)m(b)s(er)328 1783 y(\014elds)39 b Fm(Q)655 1802 y Fk(p)733 1783 y Fq(w)m(e)h(ha)m(v)m(e:)58 b(\001)1240 1811 y Fm(Q)1311 1830 y Fg(p)1390 1783 y Fq(=)p Fn(j)38 b(\001)g(j)1626 1798 y Fk(p)1704 1783 y Fq(,)i Fo(p)f Fq(is)g(a)f(prime.)62 b(Moreo)m(v)m(er)40 b Fn(j)e Fo(z)43 b Fn(j)3001 1747 y Fe(2)3001 1823 y Fm(C)3071 1783 y Fq(=)38 b(\001)3266 1811 y Fm(C)3336 1783 y Fq(\()p Fo(z)t Fq(\)\))i(,)328 1931 y Fn(j)33 b Fo(h)g Fn(j)506 1895 y Fe(4)506 1955 y Fk(H)573 1931 y Fq(=)g(\001)763 1946 y Fk(H)831 1931 y Fq(\()p Fo(h)p Fq(\))i(and)h(generally)f(:)50 b(\001)1767 1946 y Fk(K)1836 1931 y Fq(\()p Fo(k)s Fq(\))33 b(=)p Fn(j)f Fo(k)37 b Fn(j)2251 1895 y Fk(d)2251 1955 y(K)2319 1931 y Fo(;)17 b(d)32 b Fn(\025)i Fq(1)p Fo(;)17 b(d)32 b Fn(2)i Fm(N)48 b Fq(\(cf.[W)-8 b(e,I.2)37 b(and)328 2051 y(Corollary)31 b(2]\).)474 2172 y(In)i(the)g(sequel)h(w)m(e)f (rather)g(prefer)g(to)f(w)m(ork)i(with)e(\001)2431 2187 y Fk(K)2500 2172 y Fq(\()p Fo(k)s Fq(\))g(instead)h(of)f Fn(j)27 b Fo(k)k Fn(j)3275 2187 y Fk(K)3343 2172 y Fq(.)474 2292 y(W)-8 b(e)28 b(start)e(from)g(the)h(follo)m(wing)e(simple)g (algebraically-measure)e(form)m(ulas)j(for)g(the)328 2412 y(Haar)35 b(measure)g Fo(H)1035 2427 y Fk(K)1104 2412 y Fq(,)h(\(where)g(only)f(algebraic)e(and)j(measure)f(structure)i (are)e(imp)s(or-)328 2533 y(tan)m(t\).)43 b(That)33 b(is)f(an)h(short)g (algebraic)e(calculus.)328 2722 y Fr(Lemma)37 b(4)49 b Fb(L)-5 b(et)30 b Fo(K)37 b Fb(b)-5 b(e)30 b(a)g(lo)-5 b(c)g(al)29 b(\014eld.)43 b(Then)29 b(for)h(e)-5 b(ach)29 b Fr(in)m(tegrable)g Fb(function)h Fo(f)41 b Fb(w.r.t)328 2842 y Fo(H)409 2857 y Fk(K)474 2838 y Fd(\003)548 2842 y Fb(on)34 b Fo(K)777 2806 y Fl(\003)852 2842 y Fb(we)g(have:)988 3091 y Fq(\()p Fo(i)p Fq(\))1141 2974 y Ff(Z)1187 3162 y Fk(K)1252 3143 y Fd(\003)1308 3091 y Fo(f)11 b Fq(\()p Fo(\025)1462 3050 y Fl(\000)p Fe(1)1556 3091 y Fq(\))1604 3023 y Fo(dH)1736 3038 y Fk(K)1804 3023 y Fq(\()p Fo(\025)p Fq(\))p 1604 3067 333 4 v 1629 3159 a(\001)1710 3174 y Fk(K)1779 3159 y Fq(\()p Fo(\025)p Fq(\))2037 3091 y(=)2168 2974 y Ff(Z)2214 3162 y Fk(K)2279 3143 y Fd(\003)2335 3091 y Fo(f)g Fq(\()p Fo(\025)p Fq(\))2537 3023 y Fo(dH)2669 3038 y Fk(K)2736 3023 y Fq(\()p Fo(\025)p Fq(\))p 2537 3067 V 2562 3159 a(\001)2643 3174 y Fk(K)2711 3159 y Fq(\()p Fo(\025)p Fq(\))2879 3091 y Fo(;)411 b Fq(\(4.73\))328 3372 y Fb(,)35 b(i.e.)582 3325 y Fk(dH)676 3336 y Fg(K)735 3325 y Fe(\()p Fk(\025)p Fe(\))p 582 3348 250 4 v 600 3406 a(\001)659 3417 y Fg(K)717 3406 y Fe(\()p Fk(\025)p Fe(\))876 3372 y Fb(is)f(the)h Fr(Bogolub)s(o\013-Krilo\013)h(measure)g Fb(of)e Fo(I)2657 3387 y Fk(K)2726 3372 y Fb(,)g(and)1000 3656 y Fq(\()p Fo(ii)p Fq(\))1187 3539 y Ff(Z)1233 3728 y Fk(K)1298 3709 y Fd(\003)1354 3656 y Fo(f)11 b Fq(\()p Fo(\025)p Fq(\))p Fo(dH)1678 3671 y Fk(K)1745 3656 y Fq(\()p Fo(\025)p Fq(\))55 b(=)2065 3539 y Ff(Z)2111 3728 y Fk(K)2176 3709 y Fd(\003)2287 3589 y Fo(f)11 b Fq(\()p Fo(\025)p Fq(\))p 2242 3633 283 4 v 2242 3724 a(\001)2323 3690 y Fe(2)2323 3749 y Fk(K)2391 3724 y Fq(\()p Fo(\025)p Fq(\))2534 3656 y Fo(dH)2666 3671 y Fk(K)2734 3656 y Fq(\()p Fo(\025)p Fq(\))p Fo(:)423 b Fq(\(4.74\))328 3937 y Fr(Pro)s(of)p Fq(.)51 b(Let's)36 b(consider)g(the)g(Haar)f(measure)2111 3890 y Fk(dH)2205 3901 y Fg(K)2264 3890 y Fe(\()p Fk(\025)p Fe(\))p 2111 3914 250 4 v 2129 3971 a(\001)2188 3982 y Fg(K)2246 3971 y Fe(\()p Fk(\025)p Fe(\))2405 3937 y Fq(and)h(the)g(second)g(order)g (auto-)328 4092 y(morphism)21 b Fo(I)817 4107 y Fk(K)908 4092 y Fq(with)h Fo(I)1171 4056 y Fe(2)1163 4117 y Fk(K)1228 4098 y Fd(\003)1295 4092 y Fq(=)27 b Fo(id)1482 4107 y Fk(K)1547 4089 y Fd(\003)1587 4092 y Fq(.)40 b(Since,)25 b(ob)m(viously)d Fo(I)2400 4056 y Fl(\003)2392 4117 y Fk(K)2460 4092 y Fq(\()2508 4046 y Fk(dH)2602 4057 y Fg(K)2661 4046 y Fe(\()p Fk(\025)p Fe(\))p 2508 4069 V 2526 4127 a(\001)2585 4138 y Fg(K)2644 4127 y Fe(\()p Fk(\025)p Fe(\))2767 4092 y Fq(\))h(is)f(a)g(Haar)g(measure)328 4213 y(on)k Fo(K)547 4177 y Fl(\003)587 4213 y Fq(,)i(then)f(according) f(to)g(the)h(v)m(on)g(Neumann-W)-8 b(eil)25 b(theorem)h(there)h(exists) g(unique)328 4333 y(constan)m(t)33 b Fo(c)28 b Fq(=)g Fo(mod)1078 4348 y Fk(K)1143 4329 y Fd(\003)1182 4333 y Fq(\()p Fo(I)8 b Fq(\))32 b(with)g(the)h(prop)s(ert)m(y)1225 4592 y Fo(I)1276 4551 y Fl(\003)1315 4592 y Fq(\()1363 4524 y Fo(dH)1495 4539 y Fk(K)1563 4524 y Fq(\()p Fo(\025)p Fq(\))p 1363 4568 333 4 v 1388 4660 a(\001)1469 4675 y Fk(K)1538 4660 y Fq(\()p Fo(\025)p Fq(\))1706 4592 y(\))55 b(=)g Fo(mod)2113 4607 y Fk(K)2178 4588 y Fd(\003)2218 4592 y Fq(\()p Fo(I)8 b Fq(\))2355 4524 y Fo(dH)2487 4539 y Fk(K)2554 4524 y Fq(\()p Fo(\025)p Fq(\))p 2355 4568 V 2379 4660 a(\001)2460 4675 y Fk(K)2529 4660 y Fq(\()p Fo(\025)p Fq(\))2697 4592 y Fo(:)328 4844 y Fq(But)786 4965 y Fo(c)828 4924 y Fe(2)922 4965 y Fq(=)56 b Fo(mod)1237 4924 y Fe(2)1237 4989 y Fk(K)1302 4970 y Fd(\003)1341 4965 y Fq(\()p Fo(I)8 b Fq(\))55 b(=)g Fo(mod)1837 4980 y Fk(K)1902 4961 y Fd(\003)1941 4965 y Fq(\()p Fo(I)2030 4924 y Fe(2)2070 4965 y Fq(\))g(=)g Fo(mod)2477 4980 y Fk(K)2542 4961 y Fd(\003)2581 4965 y Fq(\()p Fo(id)2703 4980 y Fk(K)2768 4961 y Fd(\003)2808 4965 y Fq(\))g(=)g(1)p Fo(:)1898 5214 y Fq(27)p eop %%Page: 28 28 28 27 bop 328 631 a Fq(Therefore)963 664 y Ff(Z)1009 853 y Fk(K)1074 834 y Fd(\003)1130 781 y Fo(f)11 b Fq(\()p Fo(\025)1284 740 y Fl(\000)p Fe(1)1378 781 y Fq(\))1426 714 y Fo(dH)1558 729 y Fk(K)1625 714 y Fq(\()p Fo(\025)p Fq(\))p 1426 758 333 4 v 1451 850 a(\001)1532 865 y Fk(K)1600 850 y Fq(\()p Fo(\025)p Fq(\))1796 781 y(=)1899 664 y Ff(Z)1946 853 y Fk(K)2011 834 y Fd(\003)2066 781 y Fo(f)g Fq(\()p Fo(\025)p Fq(\))p Fo(dI)2360 740 y Fl(\003)2399 781 y Fq(\()2447 714 y Fo(dH)2579 729 y Fk(K)2647 714 y Fq(\()p Fo(\025)p Fq(\))p 2447 758 V 2472 850 a(\001)2553 865 y Fk(K)2622 850 y Fq(\()p Fo(\025)p Fq(\))2789 781 y(\))28 b(=)1526 1079 y(=)1630 962 y Ff(Z)1676 1150 y Fk(K)1741 1131 y Fd(\003)1797 1079 y Fo(f)11 b Fq(\()p Fo(\025)p Fq(\))1999 1011 y Fo(dH)2131 1026 y Fk(K)2198 1011 y Fq(\()p Fo(\025)p Fq(\))p 1999 1055 V 2023 1147 a(\001)2104 1162 y Fk(K)2173 1147 y Fq(\()p Fo(\025)p Fq(\))2341 1079 y Fo(:)328 1309 y Fq(\(In)39 b(fact)f(more)g(general)h (fact)f(is)g(true:)56 b(eac)m(h)40 b(left)e(\(righ)m(t\))g(Haar)g (measure)h Fo(H)3278 1324 y Fk(G)3375 1309 y Fq(on)g(a)328 1429 y(unimo)s(dular)27 b(group)j Fo(G)g Fq(is)f(sim)m(ultaneously)g (the)h(Bogolub)s(o\013-Krilo\013)c(measure)k(of)f(the)328 1549 y(in)m(v)m(ersion)k(of)f(that)g(group,)h(cf.e.g.)44 b([BR,)32 b(I)s(I.3,)h(Prop)s(osition)e(3]\).)474 1670 y(The)36 b(whole)f(ab)s(o)m(v)m(e)h(calculus)f(is)g(written)g(in)f (fact)h(on)g(the)h(page)f(174)g(in)f(the)i(1972)328 1790 y(Russian)28 b(edition)e(of)h(the)i(W)-8 b(eil's)27 b(b)s(o)s(ok)g([W) -8 b(e])28 b(\(in)f(the)i(case)f(of)g(idele)f(group)g(of)h(a)f(global) 328 1911 y(\014eld\).)474 2031 y(T)-8 b(o)31 b(obtain)f(the)h(second)h (form)m(ula)d(it)g(su\016ces)k(to)e(apply)f(the)h(\014rst)g(form)m(ula) e(for)h(the)328 2151 y("new")j(function)f Fo(\036)p Fq(\()p Fo(\025)p Fq(\))27 b(=)h Fo(f)11 b Fq(\()p Fo(\025)1484 2115 y Fl(\000)p Fe(1)1578 2151 y Fq(\))p Fo(=)p Fq(\001)1746 2166 y Fk(K)1814 2151 y Fq(\()p Fo(\025)p Fq(\).)328 2379 y Fr(Remark)37 b(5)49 b Fb(L)-5 b(et's)48 b(observe)e(that)j(the)e (formulas)g(\(4.73\))g(and)g(\(4.74\))g(in)h(the)f(c)-5 b(ase)328 2499 y(of)46 b Fo(K)57 b Fq(=)49 b Fm(R)55 b Fb(ar)-5 b(e)46 b(usual)5 b(ly)47 b(analytic)-5 b(aly)46 b(pr)-5 b(oven)46 b(,)j(by)e(using)f(1-dimensional)e(Jac)-5 b(obi)328 2620 y(formula.)474 2847 y Fq(Let's)26 b(consider)e(no)m(w)i Fo(K)7 b Fr(-segmen)m(t)28 b(\(ring\))23 b Fo(R)2209 2862 y Fk(K)2277 2847 y Fq(\()p Fo(M)5 b(;)17 b(N)10 b Fq(\))28 b(:=)g Fn(f)p Fo(k)j Fn(2)d Fo(K)34 b Fq(:)28 b Fo(M)39 b Fn(\024j)28 b Fo(k)i Fn(\024)328 2967 y Fo(N)10 b Fn(g)p Fo(;)17 b(N)38 b(>)28 b Fq(0.)42 b(Ob)m(viously)30 b Fo(R)1378 2982 y Fk(K)1474 2967 y Fq(=)e Fo(R)1652 2982 y Fk(K)1720 2967 y Fq(\(0)p Fo(;)17 b Fq(1\))29 b(Then)i(w)m(e)g(can)f(consider)g(a)f(top)s(ologically-)328 3088 y(dynamical)48 b(system)j Fo(D)1236 3103 y Fk(K)1305 3088 y Fq(\()p Fo(M)5 b(;)17 b(N)10 b Fq(\))57 b(:=)h(\()p Fo(R)1942 3103 y Fk(K)2010 3088 y Fq(\()p Fo(M)5 b(;)17 b(N)10 b Fq(\))35 b Fn(\000)f Fo(R)2537 3103 y Fk(K)2606 3088 y Fq(\()p Fo(N)2732 3052 y Fl(\000)p Fe(1)2827 3088 y Fo(;)17 b(M)2975 3052 y Fl(\000)p Fe(1)3069 3088 y Fq(\))p Fo(;)g(I)3194 3103 y Fk(K)3262 3088 y Fq(\))p Fo(;)g(N)68 b(>)328 3208 y(M)38 b(>)28 b Fq(0.)328 3411 y Fr(Lemma)37 b(5)49 b Fb(L)-5 b(et)36 b Fo(K)43 b Fb(b)-5 b(e)36 b(a)g(lo)-5 b(c)g(al)35 b(\014eld)g(and)h Fo(H)2034 3426 y Fk(K)2138 3411 y Fb(b)-5 b(e)36 b(a)f(left)h(Haar)h(me)-5 b(asur)g(e)35 b(of)h Fq(\()p Fo(K)r(;)17 b Fq(+\))p Fb(.)328 3531 y(Then)34 b(the)h(formula)1367 3701 y Fo(\026)1426 3660 y Fk(K)1426 3726 y(B)s(K)1551 3701 y Fq(\()p Fo(B)5 b Fq(\))55 b(:=)1919 3584 y Ff(Z)1965 3773 y Fk(B)2132 3634 y Fo(dH)2264 3649 y Fk(K)2332 3634 y Fq(\()p Fo(\025)p Fq(\))p 2053 3678 493 4 v 2053 3769 a(\001)2134 3784 y Fk(K)2202 3769 y Fq(\(1)22 b Fn(\000)h Fo(\025)2468 3741 y Fe(2)2507 3769 y Fq(\))2555 3701 y Fo(;)735 b Fq(\(4.75\))328 3935 y Fb(,)32 b(wher)-5 b(e)32 b Fo(B)37 b Fb(is)32 b(a)g(Bor)-5 b(el)31 b(set)h(in)g Fo(X)1561 3950 y Fk(K)1630 3935 y Fq(\()p Fo(M)5 b(;)17 b(N)10 b Fq(\))28 b(:=)f Fo(R)2169 3950 y Fk(K)2238 3935 y Fq(\()p Fo(N)5 b(;)17 b(M)10 b Fq(\))16 b Fn(\000)g Fo(R)2728 3950 y Fk(K)2797 3935 y Fq(\()p Fo(M)2939 3899 y Fl(\000)p Fe(1)3035 3935 y Fo(;)h(N)3167 3899 y Fl(\000)p Fe(1)3261 3935 y Fq(\))p Fb(,)33 b(gives)328 4056 y(a)i(Bo)-5 b(golub)g (o\013-Krilo\013)32 b(me)-5 b(asur)g(e)35 b(of)f(the)h(dynamic)-5 b(al)34 b(systewm)h Fo(D)2777 4071 y Fk(K)2845 4056 y Fq(\()p Fo(M)5 b(;)17 b(N)10 b Fq(\))p Fb(.)328 4258 y Fr(Pro)s(of)p Fq(.)43 b(It)33 b(su\016ces)h(to)f(apply)f(the)h(form)m (ula)e(\(4.74\))g(for)i(the)g(function)1003 4478 y Fo(f)11 b Fq(\()p Fo(\025)p Fq(\))55 b(=)g(\001)1462 4493 y Fk(K)1531 4478 y Fq(\(\(1)22 b Fn(\000)g Fo(\025)1834 4437 y Fe(2)1874 4478 y Fq(\))1912 4437 y Fl(\000)p Fe(1)2006 4478 y Fq(\))p Fo(\037)2105 4493 y Fk(X)2163 4504 y Fg(K)2222 4493 y Fe(\()p Fk(M)s(;N)7 b Fe(\))2434 4478 y Fq(\()p Fo(\025)p Fq(\))28 b(;)17 b Fo(\025)27 b Fn(2)h Fo(K)2907 4437 y Fl(\003)2947 4478 y Fo(:)328 4697 y Fr(Remark)37 b(6)49 b Fb(L)-5 b(et's)36 b(observe)f(that)i(Bo)-5 b(golub)g(o\013-Krilo\013) 34 b(me)-5 b(asur)g(es)35 b(ar)-5 b(e)36 b(gener)-5 b(al)5 b(ly)36 b(not)328 4817 y(uniqne.)61 b(F)-7 b(or)39 b(example)g(the)h (dynamic)-5 b(al)40 b(system)g Fo(D)2306 4832 y Fk(K)2374 4817 y Fq(\()p Fo(M)5 b(;)17 b(N)10 b Fq(\))41 b Fb(has)f(got)g(at)h (le)-5 b(ast)40 b(two)328 4947 y(invariant)c(me)-5 b(asur)g(es)35 b(:)1303 4900 y Fk(dH)1397 4911 y Fg(K)1456 4900 y Fe(\()p Fk(\025)p Fe(\))p 1258 4924 339 4 v 1258 4981 a(\001)1317 4992 y Fg(K)1375 4981 y Fe(\(1)p Fl(\000)p Fk(\025)1533 4962 y Fj(2)1569 4981 y Fe(\))1643 4947 y Fb(and)1843 4900 y Fk(dH)1937 4911 y Fg(K)1996 4900 y Fe(\()p Fk(\025)p Fe(\))p 1843 4924 250 4 v 1861 4981 a(\001)1920 4992 y Fg(K)1979 4981 y Fe(\()p Fk(\025)p Fe(\))2102 4947 y Fb(.)49 b(F)-7 b(or)36 b(the)-5 b(or)g(ems)36 b(which)f(appr)-5 b(oximate)1898 5214 y Fq(28)p eop %%Page: 29 29 29 28 bop 328 631 a Fb(the)27 b(numb)-5 b(er)26 b(of)g(invariant)g(me) -5 b(asur)g(es)26 b(of)h(a)f(dynamic)-5 b(al)26 b(system)h(,cf.)41 b([FKS,V.2,)28 b(Th.1])328 751 y(.)474 872 y(It)41 b(is)g(also)f(c)-5 b(onvenient)40 b(to)h(c)-5 b(al)5 b(l)41 b(the)g(density)2220 818 y Fk(d\026)2298 795 y Fg(K)2298 842 y(B)r(K)p 2220 848 186 4 v 2236 906 a Fk(dH)2330 917 y Fg(K)2456 872 y Fb(the)g Fo(K)7 b Fr(-P)m(oincare)45 b(metric)p Fb(,)328 992 y(sinc)-5 b(e)34 b(that)h(function)g(is)f(mo)-5 b(dele)g(d)34 b(on)h(the)f(true)i(Poinc)-5 b(ar)g(e)34 b(metric)1447 1251 y Fo(P)14 b Fq(\()p Fo(z)t Fq(\))56 b(=)2131 1184 y(1)p 1846 1228 618 4 v 1846 1319 a Fn(j)27 b Fq(1)p Fn(\000)h(j)g Fo(z)t(mid)2329 1291 y Fe(2)2397 1319 y Fn(j)2425 1291 y Fe(2)328 1527 y Fb(of)35 b Fo(R)517 1555 y Fm(C)587 1527 y Fb(.)46 b(In)35 b(the)g(numb)-5 b(er)35 b(\014eld)g Fm(C)59 b Fb(c)-5 b(ase)34 b(additional)5 b(ly)35 b(app)-5 b(e)g(ars)34 b(the)i(mo)-5 b(dule)34 b(\(cf.)46 b([Kr,)328 1647 y(I.1.4]\))474 1851 y Fq(A)35 b(quite)f(unexp)s(ected)j(for)d(us)h(w)m(as)h(a)e(help)g(from)f(the)i (h)m(uge)h Fr(p)s(oten)m(tial)i(theory)p Fq(.)328 1971 y(According)j(to)h(one)g(of)f(the)h(famous)f(sp)s(ecialists)f(in)h(the) h(p)s(oten)m(tial)e(theory)i(-)g(W.K.)328 2091 y(Ha)m(yman)h(\(cf.[HK,) g(An)g(In)m(tro)s(duction]\))f(it)g(is)h(in)f(fact,)j(a)e(theory)g(of)f (subharmonic)328 2212 y(functions,)33 b(with)f(its)g(deep)h(and)g (di\016cult)f(results:)328 2332 y(\(1\))23 b Fr(Riesz)k(theorem)c Fq(-)h(on)f(the)i(in)m(tegral)d(represen)m(tation)j(of)e(subharmonic)h (functions)328 2453 y(\()34 b(i.e.)49 b(existence)37 b(of)d(an)g Fr(Abstract)39 b(Ho)s(dge)h(Decomp)s(osition)33 b Fq(of)h(a)g(subharmonic)328 2573 y(function)29 b Fo(s)p Fq(\()p Fo(x)p Fq(\))g(in)g(a)g(domain)f(from)g Fm(R)1738 2537 y Fk(m)1834 2573 y Fq(with)h(resp)s(ect)i(to)e(the)h Fr(P)m(oisson)j(t)m(yp)s(e)c Fq(k)m(ernel)328 2693 y Fo(K)411 2708 y Fk(m)478 2693 y Fq(\()p Fo(x)i Fn(\000)g Fo(y)t Fq(\),)47 b(whic)m(h)f(is)f(moreo)m(v)m(er)h(Wiener-Hopf)e(t)m (yp)s(e)j(and)e Fr(algebraic)f Fq(\):)69 b(there)328 2814 y(exists)29 b(a)f Fr(Riesz)k(measure)d Fo(dR)f Fq(=)g Fn(5)1740 2778 y Fe(2)1779 2814 y Fo(sdx)g Fq(and)h Fr(harmonic)f Fo(h)p Fq(\()p Fo(x)p Fq(\))g(outside)h(a)f(comp)s(ct)328 2934 y(set)33 b Fo(E)39 b Fq(function)32 b(with)1003 3154 y(\()p Fo(R)q(T)14 b Fq(\))83 b Fo(s)p Fq(\()p Fo(x)p Fq(\))55 b(=)1672 3037 y Ff(Z)1718 3226 y Fk(E)1794 3154 y Fo(K)1877 3169 y Fk(m)1944 3154 y Fq(\()p Fo(x)22 b Fn(\000)h Fo(y)t Fq(\))p Fo(dR)q Fq(\()p Fo(y)t Fq(\))47 b(+)j Fo(h)p Fq(\()p Fo(x)p Fq(\))p Fo(;)427 b Fq(\(4.76\))328 3408 y(\(cf.[HK,3.5,)26 b(Theorem)g(3.9]\).)40 b(Ob)m(viously)25 b(the)h(Riesz)f(measure)g(pla)m(ys)g(the)h(role)e(of)g(the)328 3529 y(Ho)s(dge)36 b(measure)g(in)g(this)f(Abstract)i(Ho)s(dge)f (Decomp)s(osition.)52 b(The)37 b(role)e(of)g Fr(Green)328 3649 y(functions)k Fo(K)896 3664 y Fk(m)963 3649 y Fq(\()p Fo(x)p Fq(\))h(,)h(whic)m(h)f(are)g(equal)f(to)h Fo(l)r(og)j Fn(jj)c Fo(x)h Fn(jj)2463 3664 y Fe(2)2541 3649 y Fq(if)f Fo(m)h Fq(=)f(2)h(and)g Fn(jj)f Fo(x)h Fn(jj)3410 3613 y Fe(2)p Fl(\000)p Fk(m)3410 3674 y(m)328 3784 y Fq(if)d Fo(m)f Fn(\025)h Fq(3)h(,)h(where)g Fn(jj)d Fo(x)h Fn(jj)1338 3799 y Fk(m)1404 3784 y Fq(:=)1543 3689 y Ff(q)p 1626 3689 326 4 v 28 x(P)1714 3744 y Fk(m)1714 3808 y(j)t Fe(=1)1857 3784 y Fo(x)1912 3750 y Fe(2)1912 3807 y Fk(j)1990 3784 y Fq(is)g(m-euklidean)f(norm)h(in)g Fm(R)3093 3747 y Fk(m)3197 3784 y Fq(is)g(not)h(to)328 3904 y(o)m(v)m(erestimate.)474 4025 y(First)33 b(of)g(all)f(they)i(are)g Fr(fundamen)m(tal)39 b(solutions)32 b Fq(of)h(the)h Fr(Laplace)40 b(op)s(erator)328 4145 y Fq(\001)28 b(=)f Fn(5)629 4109 y Fe(2)701 4145 y Fq(in)32 b Fm(R)880 4109 y Fk(m)947 4145 y Fq(:)1607 4265 y(\001)p Fo(K)1771 4280 y Fk(m)1838 4265 y Fq(\()p Fo(x)p Fq(\))56 b(=)f Fo(\016)2199 4280 y Fe(0)2239 4265 y Fq(\()p Fo(x)p Fq(\))p Fo(;)328 4440 y Fq(i.e.)40 b(they)25 b(are)f(also)f Fr(harmonic)k(functions)d Fq(in)f(the)i(domain)d Fm(R)2667 4403 y Fk(m)2738 4440 y Fn(\000)5 b(f)p Fq(0)p Fn(g)p Fq(.)40 b(Ho)m(w)m(ev)m(er,)28 b(for)328 4560 y(our)36 b(purp)s(oses)i(,)f(more)f(imp)s(ortan)m(t)f(is)h(the)h(b)s (elo)m(w)f Fr(Brelot's)k(theorem)p Fq(\(cf.[HK,3.6,)328 4680 y(Theorem)c(3.10]\))e(on)h(there)h(existence)h(of)e(the)h Fr(harmonic)j(measure)p Fq(.)52 b(Let)36 b Fo(D)i Fq(b)s(e)d(a)328 4801 y Fr(regular)26 b Fq(and)i(b)s(ounded)f(domain)f(of)g Fm(R)1791 4765 y Fk(m)1884 4801 y Fq(with)h(the)g(b)s(order)g Fo(@)5 b(D)31 b Fq(=)d Fo(F)14 b Fq(.)41 b(Then)28 b(for)f(eac)m(h)328 4921 y Fo(x)39 b Fn(2)g Fo(D)j Fq(and)d(arbitrary)f(Borel)g(set)i Fo(B)k Fq(in)38 b Fo(F)53 b Fq(,)40 b(there)g(exists)g(a)f Fr(unique)g Fq(function)f(of)1898 5214 y(29)p eop %%Page: 30 30 30 29 bop 328 631 a Fq(t)m(w)m(o)33 b(v)-5 b(ariables)31 b(:)44 b Fo(!)t Fq(\()p Fo(x;)17 b(B)5 b Fq(;)17 b Fo(D)s Fq(\))31 b(with)i(the)g(follo)m(wing)c(three)34 b(prop)s(erties)e(:)328 751 y(\(i\))39 b(for)h(eac)m(h)i Fo(x)f Fn(2)h Fo(D)s(;)17 b(!)t Fq(\()p Fo(x;)g(:)p Fq(;)g Fo(D)s Fq(\))38 b(is)i(a)g Fr(probabilit)m(y)f Fq(Borel)h(measure)g(on)h Fo(@)5 b(D)44 b Fq(,)e(i.e.)328 872 y Fo(!)t Fq(\()p Fo(x;)17 b(@)5 b(D)s Fq(;)17 b Fo(D)s Fq(\))27 b(=)h(1.)328 992 y(\(ii\)F)-8 b(or)30 b(eac)m(h)j(Borel)f(set)h Fo(B)g Fn(\032)28 b Fo(@)5 b(D)s(;)17 b(!)t Fq(\()p Fo(:;)g(B)5 b Fq(;)17 b Fo(D)s Fq(\))32 b(is)g(a)g Fr(harmonic)37 b(function)32 b Fq(on)g Fo(D)s Fq(.)328 1112 y(\(iii\))e(\(Abstract)j (Ho)s(dge)f(Decomp)s(osition)e(of)i(harmonic)g(extension\).)328 1233 y(If)g Fo(f)11 b Fq(\()p Fo(\030)5 b Fq(\))32 b(is)g(a)g Fr(semi-con)m(tin)m(uous)g Fq(function)g(on)g Fo(@)5 b(D)s Fq(,)34 b(then)f(the)g(form)m(ula)1103 1414 y(~)1082 1440 y Fo(f)10 b Fq(\()p Fo(x)p Fq(\))56 b(=)1458 1323 y Ff(Z)1504 1511 y Fk(@)t(D)1626 1440 y Fo(f)11 b Fq(\()p Fo(\030)5 b Fq(\))p Fo(d!)t Fq(\()p Fo(x;)17 b(\030)5 b Fq(;)17 b Fo(D)s Fq(\))25 b(;)44 b Fo(\030)32 b Fn(2)c Fo(D)d Fn(\000)e Fo(F)s(;)477 b Fq(\(4.77\))328 1645 y(giv)m(es)33 b(the)g(harmonic)e(extension)i(of)g Fo(f)43 b Fq(from)31 b Fo(F)46 b Fq(to)33 b Fo(D)s Fq(.)328 1766 y(The)24 b(measure)f Fo(!)t Fq(\()p Fo(x;)17 b(:)p Fq(;)g Fo(D)s Fq(\))22 b(is)g(called)g(the)h Fr(harmonic)i(measure)f Fq(in)e Fo(x)h Fq(w.r.t.)41 b Fo(D)25 b Fq(and)e(its)328 1886 y(existence)36 b(follo)m(ws)d(from)g(the)h(w)m(ell-kno)m(wn)h (Riesz)f(theorem)g(on)g(the)h(represen)m(tations)328 2007 y(of)d(linear)f(functionals)g(as)i(measures.)328 2173 y Fr(Remark)k(7)49 b Fb(The)28 b(family)g Fo(!)t Fq(\()p Fo(D)s Fq(\))f(:=)h Fn(f)p Fo(!)t Fq(\()p Fo(x;)17 b(:)p Fq(;)g Fo(D)s Fq(\))26 b(:)i Fo(x)g Fn(2)g Fo(@)5 b(D)s Fn(g)29 b Fb(of)g(harmonic)e(me)-5 b(asur)g(es)328 2293 y(on)43 b Fo(D)k Fb(solves)c(the)h(famous)f Fr(Diric)m(hlet)i (problem)e Fb(for)h(a)f(p)-5 b(air)44 b Fq(\()p Fo(D)s(;)17 b(@)5 b(D)s Fq(\))p Fb(:)63 b(\014nd)43 b(the)328 2414 y Fr(harmonic)j Fb(function)h Fo(u)p Fq(\()p Fo(x)p Fq(\))g Fb(on)g Fo(D)j Fb(and)d Fr(con)m(tin)m(uous)f Fb(on)h Fo(D)34 b Fn(\\)e Fo(@)5 b(D)51 b Fb(which)46 b(takes)328 2534 y(given)36 b(values)g(on)g Fo(F)14 b Fb(.)49 b(It)36 b(is)g(wel)5 b(l-known)35 b(that)i(the)f(solution)h(of)f(a)g(Dirichlet) g(pr)-5 b(oblem)328 2655 y(is)35 b Fr(unique)p Fb(,)g(under)f(the)h (assumption)f(that)h(one)g(exists.)474 2821 y Fq(The)h Fr(Riesz)i(measure)d Fo(R)g Fq(and)f(family)e Fo(!)t Fq(\()p Fo(D)s Fq(\))h(of)h Fr(harmonic)39 b(measures)c Fq(com-)328 2942 y(monly)30 b(app)s(ear)i(in)e(the)i(famous)f Fr(P)m(oisson-Jensen)37 b(form)m(ula)30 b Fq(\()i(a)f(double)g (Abstract)328 3062 y(Ho)s(dge)48 b(Decomp)s(osition)e(of)h(a)h Fr(subharmonic)g Fq(function)g Fo(s)p Fq(\()p Fo(x)p Fq(\))g(on)h(a)f(domain)e Fo(D)s Fq(,)328 3182 y(cf.[HK,)33 b(Sec.3.7,Th.3.14]\):)902 3389 y Fo(s)p Fq(\()p Fo(x)p Fq(\))55 b(=)1265 3272 y Ff(Z)1312 3461 y Fk(@)t(D)1433 3389 y Fo(s)p Fq(\()p Fo(\030)5 b Fq(\))p Fo(d!)t Fq(\()p Fo(x;)17 b(\030)5 b Fq(;)17 b Fo(D)s Fq(\))i Fn(\000)2189 3272 y Ff(Z)2235 3461 y Fk(D)2316 3389 y Fo(g)t Fq(\()p Fo(x;)e(\030)5 b Fq(;)17 b Fo(D)s Fq(\))p Fo(dR)q Fq(\()p Fo(\030)5 b Fq(\))p Fo(:)328 3595 y Fq(The)44 b(second)g(in)m(tegral)e (is)g(often)h(called)f(the)h Fr(Riesz)49 b(p)s(oten)m(tial)41 b Fq(from)h(the)h Fr(Green)328 3715 y(function)32 b Fo(g)t Fq(.)474 3836 y(F)-8 b(rom)22 b(the)h(ab)s(o)m(v)m(e)h(men)m(tioned)f (mac)m(hinery)g(of)f(the)i(p)s(oten)m(tial)d(theory)j(w)m(e)g(need)g (only)328 3956 y(some)35 b Fr(simple)e Fq(result)i(concerning)g(the)g (Green)g(function)g Fo(K)2615 3971 y Fe(6)2654 3956 y Fq(\()p Fo(x)p Fq(\))d(=)p Fn(jj)f Fo(x)h Fn(jj)3123 3915 y Fl(\000)p Fe(4)3123 3978 y(6)3217 3956 y Fo(;)17 b(x)32 b Fn(6)p Fq(=)f(0)k(,)328 4076 y(i.e.)43 b(the)33 b(case)g Fo(m)28 b Fq(=)g(3.)474 4197 y(Before)37 b(w)m(e)h(write)f (the)g(main)e(needed)j(form)m(ulas)e(w)m(e)i(recall)d(the)i(v)m(ery)i (imp)s(ortan)m(t)328 4317 y(notion)d(of)g(the)h Fr(Hilb)s(ert)j (transform)c Fq(on)h(a)f(lo)s(cal)e(\014eld)j(.)56 b(Th)m(us,)39 b(let)d Fo(K)44 b Fq(b)s(e)37 b(a)f(lo)s(cal)328 4438 y(\014eld)d(\(not)h(necessary)i(comm)m(utativ)m(e\).)46 b(F)-8 b(or)33 b(some)h(class)f(of)h(admissible)d(functions)j Fo(f)328 4558 y Fq(de\014ned)j(on)e(a)h(Borel)e(subset)k Fo(B)i Fq(of)35 b Fo(K)43 b Fq(w)m(e)37 b(de\014ne)f Fo(K)7 b Fr(-Hilb)s(ert)39 b(transform)d Fq(\()p Fn(H)3401 4573 y Fk(K)3469 4558 y Fo(f)11 b Fq(\))328 4678 y(of)32 b Fo(f)43 b Fq(as)33 b(the)g(in)m(tegral)1005 4905 y(\()p Fn(H)1127 4920 y Fk(K)1196 4905 y Fo(f)11 b Fq(\)\()p Fo(x)p Fq(\))55 b(:=)1637 4788 y Ff(Z)1683 4976 y Fk(B)1905 4837 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p 1771 4881 454 4 v 1771 4973 a(\001)1852 4988 y Fk(K)1920 4973 y Fq(\()p Fo(x)23 b Fn(\000)f Fo(y)t Fq(\))2234 4905 y Fo(dH)2366 4920 y Fk(K)2434 4905 y Fq(\()p Fo(y)t Fq(\);)44 b Fo(x)28 b Fn(2)g Fo(B)5 b(:)401 b Fq(\(4.78\))1898 5214 y(30)p eop %%Page: 31 31 31 30 bop 328 631 a Fq(If)39 b Fo(B)45 b Fq(is)39 b(pre-compact)g(,)j (then)e(w)m(e)g(sa)m(y)h(on)e(a)h Fr(compact)k Fo(K)7 b Fr(-Hilb)s(ert)44 b(transform)p Fq(.)328 751 y(More)f(generally)-8 b(,)44 b(i\013)e Fo(\026)g Fq(is)g(an)h(admissible)e(measure)i(on)f Fo(K)50 b Fq(,)45 b(then)f(w)m(e)g(can)f(de\014ne)328 872 y Fo(K)7 b Fq(-Hilb)s(ert)31 b(transform)g(\()p Fn(H)q Fo(\026)p Fq(\))h(of)g(a)h(measure)g Fo(\026)p Fq(:)1210 1096 y(\()p Fn(H)q Fo(\026)p Fq(\)\()p Fo(x)p Fq(\))55 b(:=)1774 979 y Ff(Z)1820 1167 y Fk(K)2024 1028 y Fo(d\026)p Fq(\()p Fo(y)t Fq(\))p 1915 1073 454 4 v 1915 1164 a(\001)1996 1179 y Fk(K)2065 1164 y Fq(\()p Fo(x)22 b Fn(\000)h Fo(y)t Fq(\))2407 1096 y(;)44 b Fo(x)28 b Fn(2)g Fo(K)r(:)550 b Fq(\(4.79\))474 1315 y(Observ)m(e)40 b(that)e(k)m(ernel)h(of)f Fo(K)7 b Fq(-Hilb)s(ert)36 b(transform)h(is)h Fr(algebraic)f Fq(and)h Fr(Wiener-)328 1436 y(Hopf)f(t)m(yp)s(e)p Fq(.)474 1556 y(In)28 b(the)g(most)e(imp)s(ortan)m(t)g(case)i(of)f Fo(K)35 b Fq(=)27 b Fm(R)36 b Fq(of)27 b(reals)g(,)h(the)g("mo)s(dule") d(do)s(es)j(not)f(ap-)328 1676 y(p)s(ear,)i(i.e.)41 b(\(cf.e.g.)i([Pi,) 28 b(Sections)h(5.3)f(and)g(5.4]\))f(and)i(the)f("true")g(Hilb)s(ert)f (transform)328 1797 y(is)32 b(giv)m(en)h(as)1328 1966 y(\()p Fn(H)1450 1995 y Fm(R)1520 1966 y Fo(f)11 b Fq(\)\()p Fo(x)p Fq(\))55 b(=)1949 1899 y(1)p 1944 1943 59 4 v 1944 2035 a Fo(\031)2030 1849 y Ff(Z)2113 1876 y Fe(+)p Fl(1)2076 2038 y(\0001)2269 1899 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)p 2269 1943 288 4 v 2299 2035 a(y)25 b Fn(\000)d Fo(x)2566 1966 y(;)328 2164 y Fq(whereas)34 b(the)f Fr(b)s(ounded)h Fq(Hilb)s(ert)d(transform)h(is)g(de\014ned)i (as)1359 2363 y(\()1427 2338 y(~)1397 2363 y Fn(H)1481 2391 y Fm(R)1551 2363 y Fo(f)11 b Fq(\)\()p Fo(x)p Fq(\))56 b(=)1981 2296 y(1)p 1976 2340 59 4 v 1976 2431 a Fo(\031)2061 2246 y Ff(Z)2144 2272 y Fe(+1)2107 2434 y Fl(\000)p Fe(1)2265 2296 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)p 2265 2340 288 4 v 2295 2431 a(y)25 b Fn(\000)e Fo(x)2563 2363 y(;)328 2582 y Fq(and)33 b(leads)f(to)g(the)h(theory)g(of)g Fr(singular)j(in)m(tegrals)31 b Fq(\(cf.e.g.)44 b([St]\).)474 2703 y(The)23 b(imp)s(ortance)d(of)1295 2678 y(~)1266 2703 y Fn(H)1350 2731 y Fm(R)1442 2703 y Fq(follo)m(ws)g(from)h(the)h (fact)g(that)f(it)g(solv)m(es)i(a)e(v)m(ery)j(imp)s(ortan)m(t)328 2823 y(F)-8 b(redholm)31 b(equation)h(of)g(the)h(\014rst)g(order:)1526 2924 y Ff(Z)1609 2951 y Fe(+1)1572 3113 y Fl(\000)p Fe(1)1730 2974 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)p 1730 3018 V 1760 3110 a(y)25 b Fn(\000)e Fo(x)2083 3042 y Fq(=)55 b Fo(g)t Fq(\()p Fo(x)p Fq(\))p Fo(;)894 b Fq(\(4.80\))328 3256 y(whic)m(h)38 b(is)f(called)f(the)h(in)m(tegral)f(equation)h(of)g (the)g Fr(pro\014le)43 b(of)g(a)g(wing)f(of)h(a)h(plane)328 3376 y Fq(\(cf.e.g.)49 b([Pi,)34 b(Sect.5.4]\),)h(whereas)h Fn(H)1755 3404 y Fm(R)1859 3376 y Fq(solv)m(es)f(a)f(v)m(ery)i(imp)s (ortan)m(t)c(F)-8 b(redholm)32 b(equa-)328 3496 y(tion)23 b(of)h(the)h(second)g(order)g(of)f(this)g(kind-)f(so)i(called)e Fr(Carleman)k(in)m(tegral)g(equation)328 3617 y Fq (\(cf.[Pi,Sect.5.5]\).)328 3776 y Fr(Remark)37 b(8)49 b Fb(Observe)44 b(additional)5 b(ly,)48 b(that)e(\(4.80\))e(is)h(an)h (analo)-5 b(gue)44 b(of)i(the)f(family)328 3897 y(of)i(the)g(imp)-5 b(ortant)48 b Fr(Ab)s(el)k(in)m(tegral)f(equations)d Fb(\(in)f(the)g(class)g(of)g(the)g Fr(V)-9 b(oltera)328 4017 y(in)m(tegral)36 b(eqations)f Fb(of)f(the)h(\014rst)g(or)-5 b(der\):)1510 4124 y Ff(Z)1593 4150 y Fk(x)1556 4313 y Fe(0)1705 4174 y Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p Fo(dy)p 1663 4218 371 4 v 1663 4310 a Fq(\()p Fo(x)23 b Fn(\000)f Fo(y)t Fq(\))1968 4281 y Fk(m)2099 4241 y Fq(=)56 b Fo(g)t Fq(\()p Fo(x)p Fq(\))p Fo(:)877 b Fq(\(4.81\))328 4466 y Fb(F)-7 b(or)43 b Fo(m)h Fb(with)g Fq(0)f Fo(<)i(m)f(<)g Fq(1)g Fb(,)i(the)d(solution)h(of)f(\(4.81\)is)g(given)g(by)h(a)f Fr(self-similar)328 4587 y Fb(formula:)1045 4756 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))56 b(=)1700 4689 y(1)p 1432 4733 585 4 v 1432 4825 a(\000\(1)22 b Fn(\000)g Fo(m)p Fq(\)\000\()p Fo(x)p Fq(\))2065 4689 y Fo(d)p 2037 4733 107 4 v 2037 4825 a(dx)2170 4639 y Ff(Z)2253 4665 y Fk(x)2216 4828 y Fe(0)2441 4689 y Fo(g)t Fq(\()p Fo(y)t Fq(\))p Fo(dy)p 2323 4733 516 4 v 2323 4825 a Fq(\()p Fo(x)h Fn(\000)f Fo(y)t Fq(\))2628 4796 y Fe(\(1)p Fl(\000)p Fk(m)p Fe(\))2849 4756 y Fo(:)328 4965 y Fb(It)35 b(c)-5 b(an)34 b(b)-5 b(e)35 b(\014nd)f(in)g(the)h(Titchmarsh)f(p)-5 b(ap)g(er)34 b([Ti].)1898 5214 y Fq(31)p eop %%Page: 32 32 32 31 bop 328 631 a Fr(Remark)37 b(9)49 b Fb(In)26 b(the)g(c)-5 b(ase)26 b(of)h Fo(p)p Fb(-adic)f(\014elds)g Fm(Q)2034 650 y Fk(p)2074 631 y Fb(,)i(the)f Fm(Q)2357 650 y Fk(p)2396 631 y Fb(-Hilb)-5 b(ert)27 b(tr)-5 b(ansforms)26 b(pr)-5 b(ob)g(ably)328 751 y(\014rst)38 b(wer)-5 b(e)37 b(c)-5 b(onsider)g(e)g(d)37 b(in)g(the)h Fr(Vladimiro)m(v)d Fb(et)j(al.)53 b(p)-5 b(ap)g(er'[VWZ])37 b(as)g(the)h Fo(\015)5 b Fb(-or)-5 b(der)328 872 y Fr(deriv)f(ativ)m(e)29 b Fo(D)922 836 y Fk(\015)966 872 y Fo(f)42 b Fb(of)30 b(a)h(lo)-5 b(c)g(al)5 b(ly)30 b(c)-5 b(onstant)31 b(function)f Fo(f)11 b Fb(.)43 b(It)31 b(is)f(describ)-5 b(able)30 b(by)g(pseudo-)328 992 y(di\013er)-5 b(ential)34 b(op)-5 b(er)g(ator)34 b(and)h(explicitly)f(written)h(as)328 1262 y Fo(D)412 1220 y Fk(\015)456 1262 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))55 b(=)833 1144 y Ff(Z)879 1333 y Fm(Q)949 1352 y Fg(p)1017 1262 y Fn(j)28 b Fo(\030)k Fn(j)1176 1220 y Fk(\015)1176 1286 y(p)1269 1235 y Fq(^)1248 1262 y Fo(f)10 b Fq(\()p Fo(\030)5 b Fq(\))p Fo(\037)1491 1277 y Fk(p)1530 1262 y Fq(\()p Fn(\000)p Fo(\030)g(x)p Fq(\))p Fo(H)1867 1277 y Fk(p)1907 1262 y Fq(\()p Fo(d\030)g Fq(\))55 b(=)27 b(=)2426 1194 y Fo(p)2475 1158 y Fk(\015)2542 1194 y Fn(\000)22 b Fq(1)p 2354 1238 409 4 v 2354 1330 a(1)f Fn(\000)i Fo(p)2573 1301 y Fl(\000)p Fk(\015)t Fl(\000)p Fe(1)2789 1144 y Ff(Z)2835 1333 y Fm(Q)2906 1352 y Fg(p)2973 1194 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))22 b Fn(\000)h Fo(f)11 b Fq(\()p Fo(y)t Fq(\))p 2973 1238 498 4 v 2985 1338 a Fn(j)27 b Fo(x)c Fn(\000)f Fo(y)31 b Fn(j)3324 1291 y Fk(\015)t Fe(+1)3324 1346 y Fk(p)3480 1262 y Fo(H)3561 1277 y Fk(p)3601 1262 y Fq(\()p Fo(dy)t Fq(\))p Fo(;)3317 1446 y Fq(\(4.82\))328 1567 y Fb(wher)-5 b(e)29 b Fo(\037)659 1582 y Fk(p)728 1567 y Fb(is)g(the)g(additive)g (char)-5 b(acter)29 b(of)g Fm(Q)1947 1586 y Fk(p)2016 1567 y Fb(and)2221 1540 y Fq(^)2199 1567 y Fo(f)11 b Fq(\()p Fo(\030)5 b Fq(\))29 b Fb(stand)g(for)g(the)g(F)-7 b(ourier)29 b(tr)-5 b(ans-)328 1687 y(formation)775 1616 y Ff(R)814 1715 y Fm(Q)884 1735 y Fg(p)941 1687 y Fo(\037)1002 1702 y Fk(p)1042 1687 y Fq(\()p Fo(\030)5 b(x)p Fq(\))p Fo(f)11 b Fq(\()p Fo(x)p Fq(\))p Fo(H)1492 1702 y Fk(p)1531 1687 y Fq(\()p Fo(dx)p Fq(\))30 b Fb(of)f(the)g(function)g Fo(f)11 b Fb(.)43 b(The)29 b(de)-5 b(ep)g(er)28 b(analysis)h(of)g(p-) 328 1827 y(adic)f(fr)-5 b(actional)27 b(di\013er)-5 b(entiation)28 b Fo(D)1673 1791 y Fk(\015)1745 1827 y Fb(is)g(given)g(in)g(the)h(Ko)-5 b(chub)g(ei')27 b(p)-5 b(ap)g(er)28 b([Ka])g(,)h(wher)-5 b(e)328 1948 y(using)30 b(Madr)-5 b(e)g(cki's)29 b(the)-5 b(or)g(em)29 b(he)h(establishe)-5 b(d)29 b(the)h(existenc)-5 b(e)29 b(of)g(a)h Fr(Kac)m(h)m(ub)s(ei-Gauss)328 2068 y(measure)38 b Fo(\026)f Fb(over)h(in\014nite-dimensional)d(\014eld)i (extensions)f Fq(\012)2720 2083 y Fk(p)2798 2068 y Fb(of)h Fm(Q)2986 2087 y Fk(p)3064 2068 y Fb(,)h(which)f(is)g(a)328 2188 y Fr(harmonic)e(measure)f Fb(for)g Fo(D)1494 2152 y Fk(\015)1571 2188 y Fb(and)g(solve)-5 b(d)32 b(p-adic)h(inte)-5 b(gr)g(al)33 b(e)-5 b(quations)34 b(of)f(a)h(pr)-5 b(o\014le)328 2309 y(of)35 b(wing)f(of)g(a)h(plane.\(cf.[Ka,Pr)-5 b(op.6]\).)474 2429 y(In)35 b(the)h(se)-5 b(quel,)36 b(it)g(wil)5 b(l)35 b(b)-5 b(e)35 b(very)h(c)-5 b(onvenient)35 b(to)h(use)g(the)f Fr(language)40 b(of)f(Hamil-)328 2550 y(ton's)k(quaternions)c Fo(H)46 b Fb(\(cf.e.g.)56 b([Si,]\)-)38 b(pr)-5 b(ob)g(ably)39 b(the)g(most)f(imp)-5 b(ortant)39 b(example)328 2670 y(of)h(a)g(lo)-5 b(c)g(al)40 b(and)f(non-c)-5 b(ommutative)39 b(\014eld.)60 b(F)-7 b(r)i(om)39 b(the)i(p)-5 b(oint)40 b(of)g(view)f(of)h(the)h(the)-5 b(ory)328 2790 y(of)33 b(ve)-5 b(ctor)34 b(sp)-5 b(ac)g(es)33 b(over)g Fm(R)9 b Fb(,)34 b(the)g(\014eld)f(of)g(quaternions)h Fo(H)41 b Fb(is)34 b(only)f(the)h(4-dimensional)328 2911 y(euclide)-5 b(an-sp)g(ac)g(e)33 b Fq(\()p Fm(R)1111 2874 y Fe(4)1151 2911 y Fo(;)17 b Fn(jjj)26 b Fo(:)i Fn(jj)1416 2926 y Fe(4)1455 2911 y Fq(\))p Fb(.)474 3031 y(F)-7 b(r)i(om)49 b(the)i(physic)-5 b(al)50 b(p)-5 b(oint)50 b(of)g(view,)k(it)c(is)g (only)h(a)f(mo)-5 b(del)50 b(of)g(the)g Fr(Einstein-)328 3151 y(Mink)m(o)m(wski)38 b(time)f(-space)p Fb(:)49 b Fo(H)37 b Fq(=)30 b Fo(T)37 b Fn(\002)24 b Fm(R)2010 3115 y Fe(3)2050 3151 y Fb(,)36 b(i.e.)48 b(the)37 b(quaternions)e Fo(h)30 b Fn(2)h Fo(H)43 b Fb(c)-5 b(an)36 b(b)-5 b(e)328 3272 y(c)g(onsider)g(e)g(d)38 b(as)h(a)g(p)-5 b(osition)38 b Fo(r)h Fq(=)c(\()p Fo(x;)17 b(y)t(;)g(z)t Fq(\))36 b Fn(2)g Fm(R)2108 3236 y Fe(3)2186 3272 y Fb(of)j(a)g Fr(particle)f Fb(at)h(a)g Fr(momen)m(tum)328 3392 y Fo(t)28 b Fn(2)g Fo(T)41 b Fq(=)28 b Fm(R)43 b Fb(,)35 b(i.e.)1316 3513 y Fo(h)56 b Fq(=)f(\()p Fo(t;)17 b(r)s Fq(\))55 b(=)g(\()p Fo(t;)17 b(x;)g(y)t(;)g(z)t Fq(\))27 b Fn(2)h Fo(H)r(:)474 3687 y Fb(Final)5 b(ly,)26 b(fr)-5 b(om)25 b(the)g Fr(algebraic)f Fb(p)-5 b(oint)25 b(of)g(view,)h(it)f(is)g(very) g(c)-5 b(onvenient)24 b(to)h(c)-5 b(onsider)328 3807 y Fo(H)42 b Fb(as)35 b(a)g(pr)-5 b(o)g(duct)35 b(of)f(the)h(c)-5 b(omplex)34 b(planes)f(:)45 b Fo(H)35 b Fq(=)28 b Fm(C)46 b Fn(\002)22 b Fm(C)i Fb(.)45 b(It)35 b(me)-5 b(ans)33 b(that)1389 4027 y Fo(H)i Fn(2)28 b Fo(h)55 b Fq(=)h(\()p Fo(z)1925 4042 y Fe(1)1964 4027 y Fo(;)17 b(z)2053 4042 y Fe(2)2093 4027 y Fq(\))55 b Fn(2)h Fm(C)46 b Fn(\002)22 b Fm(C)i Fo(:)474 4247 y Fb(Obviously,)34 b(for)h(al)5 b(l)35 b Fo(h)27 b Fq(=)h(\()p Fo(z)1513 4262 y Fe(1)1553 4247 y Fo(;)17 b(z)1642 4262 y Fe(2)1681 4247 y Fq(\))28 b(=)f(\()p Fo(t;)17 b(x;)g(y)t(;)g(z)t Fq(\))35 b Fb(we)f(have)356 4485 y Fn(j)27 b Fo(h)h Fn(j)523 4500 y Fk(H)617 4485 y Fq(=)749 4387 y Ff(q)p 832 4387 662 4 v 98 x Fn(j)f Fo(z)932 4500 y Fe(1)999 4485 y Fn(j)1027 4450 y Fe(2)1027 4526 y Fm(C)1125 4485 y Fq(+)g Fn(j)g Fo(z)1328 4500 y Fe(2)1396 4485 y Fn(j)1424 4450 y Fe(2)1424 4526 y Fm(C)1549 4485 y Fq(=)1680 4387 y Ff(q)p 1763 4387 1308 4 v 98 x Fn(j)g Fo(t)h Fn(j)1909 4450 y Fe(2)1909 4526 y Fm(R)2006 4485 y Fq(+)g Fn(j)f Fo(x)h Fn(j)2276 4450 y Fe(2)2276 4526 y Fm(R)2373 4485 y Fq(+)g Fn(j)f Fo(y)k Fn(j)2639 4450 y Fe(2)2639 4526 y Fm(R)2736 4485 y Fq(+)d Fn(j)f Fo(z)33 b Fn(j)3001 4450 y Fe(2)3001 4526 y Fm(R)3070 4485 y Fo(;)220 b Fq(\(4.83\))328 4716 y Fb(and)34 b(\(cf.e.g.)44 b([We,p.11?]\))1597 4936 y Fq(\001)1678 4951 y Fk(H)1745 4936 y Fq(\()p Fo(h)p Fq(\))83 b(=)55 b Fn(j)28 b Fo(h)g Fn(j)2259 4895 y Fe(4)2325 4936 y Fo(:)965 b Fq(\(4.84\))1898 5214 y(32)p eop %%Page: 33 33 33 32 bop 328 631 a Fq(F)-8 b(or)40 b(eac)m(h)i Fo(r)783 646 y Fe(1)823 631 y Fo(;)17 b(r)911 646 y Fe(2)992 631 y Fo(>)42 b Fq(0)p Fo(;)17 b(r)1247 646 y Fe(1)1328 631 y Fo(<)42 b(r)1490 646 y Fe(2)1570 631 y Fq(,)i(let's)d(consider)g(a)g Fr(ring)f Fq(of)h Fm(R)2755 595 y Fk(m)2863 631 y Fq(:)60 b Fo(R)3025 595 y Fk(m)3092 631 y Fq(\()p Fo(r)3174 646 y Fe(1)3213 631 y Fo(;)17 b(r)3301 646 y Fe(2)3341 631 y Fq(\))84 b(:=)328 751 y Fn(f)p Fo(x)50 b Fn(2)h Fm(R)666 715 y Fk(m)783 751 y Fq(:)f Fo(r)904 766 y Fe(1)994 751 y Fn(\024jj)g Fo(x)h Fn(jj)1339 766 y Fe(3)1378 751 y Fn(\024)f Fo(r)1549 766 y Fe(2)1589 751 y Fn(g)45 b Fq(\(with)h(the)g (radiuses)g(:inner)g Fo(r)2861 766 y Fe(1)2946 751 y Fq(and)g(outer)g Fo(r)3461 766 y Fe(2)3501 751 y Fq(\).)328 872 y(By)41 b Fo(S)555 836 y Fk(m)p Fl(\000)p Fe(1)549 896 y Fl(1)711 872 y Fq(\()p Fo(r)s Fq(\))f(:=)h Fn(f)p Fo(x)f Fn(2)h Fm(R)1336 835 y Fk(m)1443 872 y Fq(:)p Fn(jj)f Fo(x)h Fn(jj)1718 887 y Fk(m)1784 872 y Fq(=)f Fo(r)s Fn(g)g Fq(=)h Fo(R)2229 836 y Fk(m)2295 872 y Fq(\()p Fo(r)m(;)17 b(r)s Fq(\))40 b(w)m(e)h(denote)g(the)f(\()p Fo(m)28 b Fn(\000)g Fq(1\)-)328 992 y(dimensional)i Fr(sphere)j Fq(of)f(radius)h Fo(r)s Fq(.)43 b(Moreo)m(v)m(er)34 b(w)m(e)f(write)g Fo(S)2623 956 y Fk(m)2617 1017 y Fl(1)2724 992 y Fq(instead)g(of)f Fo(S)3238 956 y Fk(m)3232 1017 y Fl(1)3306 992 y Fq(\(1\).)328 1196 y Fr(Prop)s(osition)j(2)49 b Fb(\()p Fr(The)f(existence)g(of)g (the)g(-pro\014le)f(of)i(a)f(wing)f(of)h(a)h(plane)328 1316 y(equation)37 b(in)g(measures)p Fb(\))328 1436 y(Ther)-5 b(e)46 b(exists)g(such)h(a)f(pr)-5 b(ob)g(ability)46 b(Bor)-5 b(el)46 b(me)-5 b(asur)g(e)46 b Fo(R)i Fb(on)e Fo(S)2708 1400 y Fe(3)2794 1436 y Fb(that)h(for)f(e)-5 b(ach)46 b Fo(r)52 b Fn(2)328 1557 y Fo(X)t Fq(\()p Fo(M)5 b(;)17 b(N)10 b Fq(\))49 b Fb(with)f Fo(N)63 b(>)52 b(M)64 b(>)52 b Fq(0)c Fb(the)g(fol)5 b(lowing)47 b(A)n(bstr)-5 b(act)49 b(Ho)-5 b(dge)48 b(De)-5 b(c)g(omp)g(osition)328 1677 y(holds)1381 1827 y Fq(\001)1462 1786 y Fl(\000)p Fe(1)1557 1827 y Fq(\()p Fo(l)1626 1786 y Fe(2)1665 1827 y Fq(\))55 b(=)1890 1710 y Ff(Z)1936 1899 y Fk(S)1983 1880 y Fj(3)2149 1760 y Fo(dR)q Fq(\()p Fo(y)t Fq(\))p 2048 1804 456 4 v 2048 1896 a(\001\()p Fo(r)2214 1867 y Fe(2)2275 1896 y Fn(\000)23 b Fo(y)2427 1867 y Fe(2)2465 1896 y Fq(\))2513 1827 y Fo(:)777 b Fq(\(4.85\))328 2087 y Fr(Pro)s(of)p Fq(.\(1\))43 b(First)g(w)m(e)i(giv)m(e)f(a)g(pro)s(of)f (of)h(this)f(prop)s(osition)f(on)i(the)h("in)m(tuiv)m(e)e(lev)m(el")328 2207 y(based)d(on)f(the)g(Riesz)g(theorem.)63 b(Let)39 b Fo(\017)1842 2222 y Fk(n)1928 2207 y Fo(>)g Fq(0)f(b)s(e)i(arbitrary) e(sequence)j(whic)m(h)f(con-)328 2328 y(v)m(erges)34 b(to)e(zero.)44 b(Then)34 b(the)f(function)f Fn(jj)21 b(\001)h(jj)1972 2277 y Fl(\000)p Fe(\(4+)p Fk(\017)2173 2285 y Fg(n)2215 2277 y Fe(\))1972 2349 y(6)2279 2328 y Fq(is)32 b Fr(subharmonic)h Fq(as)f(a)h(suitable)328 2448 y(p)s(o)m(w)m(er)i(of)e Fr(harmonic)g Fo(K)1298 2463 y Fe(6)1371 2448 y Fq(\(cf.e.g.)47 b([HK,Sect.2.3,)35 b(Corollary)d(2]\))i(and)g(ob)m(viously)f(is)328 2568 y(not)25 b(harmonic!)40 b(Therefore,)28 b(according)d(to)g(the)h Fr(Riesz)i(theorem)p Fq(,)f(there)f(exists)g(a)f(se-)328 2689 y(quence)31 b(of)e Fr(Riesz)k(measures)c Fn(f)p Fo(R)1648 2704 y Fk(n)1695 2689 y Fn(g)h Fq(and)f(a)g(sequence)j Fn(f)p Fo(h)2546 2704 y Fk(n)2593 2689 y Fn(g)d Fq(of)g(harmonic)f (functons)328 2809 y(inside)k(of)g Fo(S)782 2773 y Fe(3)854 2809 y Fq(with)g(the)h(prop)s(ert)m(y)356 3078 y Fn(jj)27 b Fo(r)j Fn(jj)569 3028 y Fl(\000)p Fe(\(4+)p Fk(\017)770 3036 y Fg(n)812 3028 y Fe(\))569 3100 y(6)871 3078 y Fq(=)1002 2961 y Ff(Z)1048 3150 y Fk(S)1095 3131 y Fj(3)1130 3150 y Fe(\(=)p Fl(f)p Fk(h)p Fl(2)p Fk(H)5 b Fe(:)p Fl(j)p Fk(h)p Fl(j)1499 3161 y Fg(H)1555 3150 y Fe(=1)p Fl(g)p Fe(\))1802 3011 y Fo(dR)1927 3026 y Fk(n)1974 3011 y Fq(\()p Fo(x)p Fq(\))p 1739 3055 430 4 v 1739 3147 a Fn(jj)27 b Fo(r)e Fn(\000)d Fo(x)28 b Fn(jj)2129 3112 y Fe(4)2129 3169 y(6)2228 3078 y Fq(+)50 b Fo(h)2410 3093 y Fk(n)2457 3078 y Fq(\()p Fo(r)s Fq(\))p Fo(:)710 b Fq(\(4.86\))474 3380 y(Since)39 b Fo(dR)860 3395 y Fk(n)907 3380 y Fq(\()p Fo(x)p Fq(\))f(=)g Fn(5)p Fq(\()p Fn(jj)e Fo(x)j Fn(jj)1559 3329 y Fl(\000)p Fe(\(4+)p Fk(\017)1760 3337 y Fg(n)1802 3329 y Fe(\))1559 3402 y(6)1833 3380 y Fq(\))p Fo(dx)g Fq(\(cf.[HK,Section)g(3.5]\).)61 b(Therefore)39 b(the)328 3501 y(sequence)g Fn(f)p Fo(R)860 3516 y Fk(n)907 3501 y Fq(\()p Fo(S)1011 3465 y Fe(3)1050 3501 y Fq(\))p Fn(g)e Fq(is)f Fr(b)s(ounded)p Fq(,)j(i.e.)54 b Fo(R)2016 3516 y Fk(n)2064 3501 y Fq(\()p Fo(S)2168 3465 y Fe(3)2207 3501 y Fq(\))34 b Fn(\024)h Fo(A)i Fq(,)g(for)f(some)g Fo(A)f(>)f Fq(0)i(and)h(all)328 3621 y Fo(n)28 b Fn(2)g Fm(N)13 b Fq(.)474 3742 y(According)37 b(to)g(the)h Fr(F)-9 b(rostman's)43 b(theorem)p Fq(\(cf.[HK,)37 b(Theorem)g(5.3]\),)h(w)m(e) g(can)328 3862 y(c)m(ho)s(ose)26 b(a)f(subsequence)k Fn(f)p Fo(R)1372 3877 y Fk(n)1415 3885 y Fg(p)1455 3862 y Fn(g)c Fq(whic)m(h)h(is)e Fr(w)m(eakly)29 b(con)m(v)m(ergen)m(t)c Fq(to)g(a)g(limit)c(measure)328 3982 y Fo(R)402 3997 y Fl(1)513 3982 y Fq(on)35 b Fo(S)717 3946 y Fe(3)756 3982 y Fq(,)i(i.e.)53 b Fo(R)1072 3997 y Fl(1)1180 3982 y Fq(:=)33 b(\()p Fo(w)s Fq(\))p Fo(l)r(im)1614 3997 y Fk(p)p Fl(\000)-13 b(!1)1838 3982 y Fo(R)1912 3997 y Fk(n)1955 4005 y Fg(p)1995 3982 y Fq(.)53 b(In)36 b(particular,)f(w)m (e)i(obtain)d(that)i(there)328 4103 y(exists)d Fo(h)p Fq(\()p Fo(x)p Fq(\))28 b(:=)g(lim)1079 4118 y Fk(p)p Fl(!)p Fe(+)p Fl(1)1331 4103 y Fo(h)1387 4118 y Fk(n)1430 4126 y Fg(p)1470 4103 y Fq(\()p Fo(x)p Fq(\))33 b(,)g(whic)m(h)g(is)f (harmonic)f(inside)h Fo(S)2842 4067 y Fe(3)2881 4103 y Fq(\(Harnak?\))44 b(and)1234 4386 y Fn(jj)27 b Fo(r)j Fn(jj)1447 4345 y Fl(\000)p Fe(4)1447 4410 y(6)1569 4386 y Fq(=)1700 4268 y Ff(Z)1746 4457 y Fk(S)1793 4438 y Fj(3)1907 4318 y Fo(dR)2032 4333 y Fl(1)2107 4318 y Fq(\()p Fo(x)p Fq(\))p 1858 4362 V 1858 4454 a Fn(jj)d Fo(r)e Fn(\000)d Fo(x)29 b Fn(jj)2249 4420 y Fe(4)2249 4476 y(6)2347 4386 y Fq(+)50 b Fo(h)p Fq(\()p Fo(x)p Fq(\))p Fo(:)630 b Fq(\(4.87\))328 4666 y(Since)33 b Fo(int)p Fq(\()p Fo(S)813 4630 y Fe(3)853 4666 y Fq(\))27 b(=)33 b(then)g(without)f(loss)g(of)g(generalit)m(y)g(w)m(e)i(can)f(assume)g (that)f Fo(h)c Fn(\021)g Fq(0.)474 4786 y(On)g(the)h(other)f(hand,)i (according)d(to)h(the)h Fr(Brelot's)i(theorem)p Fq(\(cf.[HK,Theorem)328 4907 y(3.10]\))49 b(applied)f(to)h(the)g(triplet)f(\()p Fo(D)1728 4922 y Fe(6)1768 4907 y Fq(\()p Fo(r)s Fq(\))p Fo(;)17 b Fq(0)p Fo(;)g(S)2094 4870 y Fe(5)2132 4907 y Fq(\()p Fo(r)s Fq(\)\)-)48 b(there)i(exists)g(the)g Fr(harmonic)1898 5214 y Fq(33)p eop %%Page: 34 34 34 33 bop 328 631 a Fr(measure)31 b Fo(!)822 646 y Fk(r)860 631 y Fq(\()p Fn(\001)p Fq(\))c(:=)h Fo(!)1183 646 y Fk(r)1220 631 y Fq(\()p Fn(\001)p Fo(;)17 b Fq(0;)g Fo(D)1504 646 y Fe(6)1543 631 y Fq(\()p Fo(r)s Fq(\)\))30 b(na)g Fo(S)1933 595 y Fe(5)1973 631 y Fq(\()p Fo(r)s Fq(\))g(with)g(the)h (prop)s(ert)m(y)h(\(cf.[HK,Lemma)328 751 y(3.1]\))1141 927 y Fn(jj)27 b Fo(l)j Fn(jj)1339 886 y Fl(\000)p Fe(4)1339 951 y(6)1460 927 y Fq(=)1592 810 y Ff(Z)1638 998 y Fk(S)1685 979 y Fj(5)1719 998 y Fe(\()p Fk(r)r Fe(\))1905 859 y Fo(d!)2017 874 y Fk(r)2054 859 y Fq(\()p Fo(y)t Fq(\))p 1838 903 410 4 v 1838 995 a Fn(jj)d Fo(l)e Fn(\000)e Fo(y)30 b Fn(jj)2209 961 y Fe(4)2209 1017 y(6)2258 927 y Fq(;)100 b Fo(l)30 b Fn(2)e Fo(D)2619 942 y Fe(6)2658 927 y Fq(\()p Fo(r)s Fq(\))p Fo(:)509 b Fq(\(4.88\))328 1161 y(Consider)46 b(the)h(map)e(of)h(measure)g(spaces)i Fo(j)2031 1176 y Fe(3)2122 1161 y Fq(:)i(\()p Fo(S)2303 1125 y Fe(3)2343 1161 y Fo(;)17 b(R)2461 1176 y Fl(1)2535 1161 y Fq(\))51 b Fn(\000)-16 b(!)50 b Fq(\()p Fo(S)2939 1125 y Fe(5)2978 1161 y Fq(\()p Fo(r)s Fq(\))p Fo(;)17 b(j)3191 1125 y Fl(\003)3185 1186 y Fe(3)3230 1161 y Fq(\()p Fo(R)3342 1176 y Fl(1)3417 1161 y Fq(\)\))46 b(,)328 1282 y(where)34 b Fo(j)650 1297 y Fe(3)690 1282 y Fq(\()p Fo(x)783 1297 y Fe(1)822 1282 y Fo(;)17 b(x)921 1297 y Fe(2)961 1282 y Fo(;)g(x)1060 1297 y Fe(3)1099 1282 y Fq(\))28 b(=)g(\()p Fo(r)s(x)1409 1297 y Fe(1)1448 1282 y Fo(;)17 b(r)s(x)1594 1297 y Fe(2)1633 1282 y Fo(;)g(r)s(x)1779 1297 y Fe(3)1819 1282 y Fo(;)g Fq(0)p Fo(;)g Fq(0)p Fo(;)g Fq(0\))30 b(.Then)1181 1439 y Ff(Z)1227 1628 y Fk(S)1274 1609 y Fj(3)1470 1489 y Fo(dR)1595 1504 y Fl(1)p 1339 1533 461 4 v 1339 1625 a Fn(jj)d Fo(r)s(x)22 b Fn(\000)h Fo(l)30 b Fn(jj)1761 1591 y Fe(4)1761 1647 y(6)1865 1556 y Fq(=)1996 1439 y Ff(Z)2042 1628 y Fk(S)2089 1609 y Fj(5)2124 1628 y Fe(\()p Fk(r)r Fe(\))2243 1489 y Fo(d)p Fq(\()p Fo(j)2378 1453 y Fl(\003)2372 1514 y Fe(3)2417 1489 y Fo(R)2491 1504 y Fl(1)2566 1489 y Fq(\)\()p Fo(y)t Fq(\))p 2243 1533 488 4 v 2282 1625 a Fn(jj)d Fo(y)e Fn(\000)e Fo(l)30 b Fn(jj)2653 1591 y Fe(4)2653 1647 y(6)2741 1556 y Fo(:)549 b Fq(\(4.89\))474 1832 y(W)-8 b(e)42 b(kno)m(w)g(\(it)e(is)h(w)m(ell-kno)m(w\),)j(as)d(a)g(h)m(uge)h (consequence)i(has)e(got)e(\(p)s(o)s(ciaga)g(za)328 1953 y(soba\))33 b(fact,)f(that)g(a)h(Haar)f(measure)h(on)f(LC)h(group)f(is) g(uniquely)h(determined)f(\(up)h(to)328 2073 y(the)g(constan)m(t)g(m)m (ultiple\)-)d(there)j(existence)h(of)e(\001)h(and)g(its)f(main)f(prop)s (erties.)474 2194 y(No)m(w,)d(w)m(e)f(explore)f(the)g(strong)g(facts)g (that)g(the)g Fr(Riesz)f Fq(and)h Fr(harmonic)f Fq(measures)328 2314 y(are)37 b(unique)g(\(in)f(the)i(w)m(ell-kno)m(w)e(sense\))j(and)e (that)g(the)g(solution)e(of)i(the)g Fr(Diric)m(hlet)328 2434 y(problem)28 b Fq(has)i(alw)m(a)m(ys)g(unique)f(solution.)41 b(Th)m(us,)32 b(from)c(the)h(one-hand)h(side)f(w)m(e)h(ha)m(v)m(e)1625 2679 y Fn(jj)d Fo(l)j Fn(jj)1823 2638 y Fl(\000)p Fe(4)1823 2703 y(6)1945 2679 y Fq(=)2076 2562 y Ff(Z)2122 2750 y Fk(S)2169 2731 y Fj(5)2203 2750 y Fe(\()p Fk(r)r Fe(\))474 2954 y Fq(W)-8 b(e)24 b(extend)h(eac)m(h)g(measure)e Fo(!)1583 2969 y Fk(n)1654 2954 y Fq(to)g(the)h(measure)53 b(~)-78 b Fo(!)2355 2969 y Fk(n)2425 2954 y Fq(on)23 b Fo(D)2635 2918 y Fe(6)2632 2978 y(1)2674 2954 y Fn(f)p Fq(0)p Fn(g)g Fq(b)m(y)i(the)f(restriction)328 3074 y(pro)s(cedure:) 1329 3194 y(~)-79 b Fo(!)1360 3209 y Fk(n)1407 3194 y Fq(\()p Fo(B)5 b Fq(\))55 b(:=)g Fo(!)1836 3209 y Fk(n)1883 3194 y Fq(\()p Fo(B)27 b Fn(\\)c Fq(\()p Fo(D)2233 3153 y Fe(6)2230 3219 y Fk(n)2299 3194 y Fn(\000)g(f)p Fq(0)p Fn(g)p Fq(\)\))p Fo(;)328 3369 y Fq(if)31 b Fo(B)38 b Fq(is)32 b(a)g(Borel)g(set)h(in)f Fr(precompact)37 b Fo(D)1923 3333 y Fe(6)1920 3393 y Fk(n)1989 3369 y Fn(\000)23 b(f)p Fq(0)p Fn(g)p Fq(.)474 3489 y(Since)38 b Fo(!)795 3504 y Fk(n)841 3489 y Fq(\()p Fo(@)5 b(D)1019 3453 y Fe(6)1016 3514 y Fk(n)1064 3489 y Fq(\))35 b(=)h(1)g(for)h(eac)m(h)h Fo(n)p Fq(,)h(the)e(sequence)j Fn(f)29 b Fq(~)-78 b Fo(!)2528 3504 y Fk(n)2574 3489 y Fn(g)37 b Fq(of)g(measures)h(on)f Fo(D)3424 3453 y Fe(6)3421 3514 y(1)3488 3489 y Fn(\000)328 3610 y(f)p Fq(0)p Fn(g)43 b Fq(is)h Fr(b)s(ounded)p Fq(.)79 b(According)44 b(to)g(the)g Fr(F)-9 b(rostman)51 b(theorem)p Fq(\(cf.[HK,Th.5.3.]\))328 3730 y(\(in)33 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b(=)569 4866 y Fo(@)5 b(B)704 4825 y Fe(4)766 4866 y Fn(\002)23 b Fq([1)f Fn(\000)g Fq(1)p Fo(=n;)17 b Fq(1)22 b(+)g(1)p Fo(=n)p Fq(])1615 4825 y Fe(2)1676 4866 y Fn([)h Fo(B)1844 4825 y Fe(4)1906 4866 y Fn(\002)f Fo(@)5 b Fq([1)23 b Fn(\000)g Fq(1)p Fo(=n;)17 b Fq(1)k(+)h(1)p Fo(=n)p Fq(])2811 4825 y Fe(2)2873 4866 y Fn([)g Fo(S)3027 4825 y Fe(5)3066 4866 y Fq(\(1)p Fo(=n)p Fq(\))p Fo(:)1898 5214 y Fq(34)p eop %%Page: 35 35 35 34 bop 328 631 a Fq(Th)m(us,)38 b(w)m(e)e(see)h(that)e(there)i (exists)f(a)f(natural)f Fr(pro)6 b(jectiv)m(e)41 b(system)f (\(sequence\))c Fq(:)328 751 y Fn(f)p Fo(@)5 b(D)518 715 y Fe(6)515 776 y Fk(n)563 751 y Fo(;)17 b(p)656 766 y Fk(mn)765 751 y Fn(g)p Fq(,)32 b(where)1723 872 y Fo(p)1772 887 y Fk(mn)1881 872 y Fq(\()p Fo(x)p Fq(\))56 b(=)328 1043 y(whic)m(h)34 b(pro)5 b(jectiv)m(e)35 b(limit)c(is)i(homeomorphic) f(with)i(the)g(3-dimensional)d(sphere)k Fo(S)3437 1006 y Fe(3)3499 1043 y Fn(\\)328 1163 y(f)p Fq(0)p Fn(g)p Fq(.)474 1283 y(Moreo)m(v)m(er,)57 b(with)50 b(the)h(ab)s(o)m(v)m(e)g (pro)5 b(jectiv)m(e)52 b(sequence)h(of)d(compact)g(top)s(ological)328 1404 y(spaces)c(is)d(asso)s(ciated)h(the)g Fr(pro)6 b(jectiv)m(e)50 b(seqence)i(of)e(measures)45 b Fq(:)66 b Fn(f)p Fo(!)3196 1419 y Fk(n)3243 1404 y Fo(;)17 b(p)3336 1419 y Fk(mn)3445 1404 y Fn(g)44 b Fq(.)328 1524 y(According)38 b(to)f(the)h Fr(Prohoro)m(v-Kisynski)43 b(theorem)37 b Fq(\(cf.e.g.)59 b([Mau]\),)40 b(there)e(ex-)328 1644 y(ists)32 b(the)h Fr(pro)6 b(jectiv)m(e)37 b(limit)d(of)k(measures)j Fq(~)-57 b Fo(!)s Fq(.)474 1765 y(Let's)25 b(consider)f(a)f(sequence)j(of)e (functions)f Fn(f)p Fo(f)2167 1729 y Fk(m)2156 1789 y(n)2233 1765 y Fn(g)h Fq(on)f Fo(D)2517 1729 y Fe(6)2514 1789 y(1)2561 1765 y Fn(\000)t(f)p Fq(0)p Fn(g)g Fq(with)g(the)h(follo)m (wing)328 1885 y(three)33 b(prop)s(erties:)328 2006 y(\(1\).)43 b Fo(f)582 1969 y Fk(m)571 2030 y(n)648 2006 y Fq(\()p Fo(x)p Fq(\))28 b(=)g(1)k(if)g Fo(x)c Fn(2)g Fo(@)5 b(D)1399 1969 y Fe(6)1396 2030 y Fk(n)1444 2006 y Fq(,)328 2126 y(\(2\).)43 b(0)27 b Fn(\024)i Fo(f)764 2090 y Fk(m)753 2151 y(n)858 2126 y Fn(\024)f Fq(1)k(on)h Fo(D)1264 2090 y Fe(6)1261 2151 y(1)1325 2126 y Fn(\000)22 b(f)p Fq(0)p Fn(g)p Fq(,)32 b(and)h Fo(f)1881 2090 y Fk(m)1870 2151 y(n)1980 2126 y Fq(is)f(con)m(tin)m(uous,)328 2246 y(\(3\).)73 b Fo(;)17 b Fq(lim)732 2261 y Fk(m)815 2246 y Fo(f)874 2210 y Fk(m)863 2271 y(n)985 2246 y Fq(=)45 b Fo(\037)1167 2261 y Fk(@)t(D)1266 2269 y Fg(n)1329 2246 y Fq(lim)1465 2261 y Fk(n;m)1610 2246 y Fo(f)1669 2210 y Fk(m)1658 2271 y(n)1735 2246 y Fq(\()p Fo(x)p Fq(\))g(=)g Fo(\037)2093 2263 y Fk(S)2140 2244 y Fj(3)2178 2246 y Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(x)45 b Fn(2)g Fo(D)2648 2210 y Fe(6)2645 2271 y(1)2688 2210 y Fl(\003)2769 2246 y Fq(,)h(where)d Fo(\037)3194 2263 y Fk(S)3241 2244 y Fj(3)3322 2246 y Fq(is)f(the)328 2367 y(c)m(haracteristic)32 b(function)g(of)g Fo(S)1486 2331 y Fe(3)1526 2367 y Fq(.)328 2487 y(Shortly)-8 b(,)32 b Fo(f)739 2502 y Fk(n)786 2487 y Fq(\()p Fo(x)p Fq(\))h(is)f(a)g("con)m(tin)m(uous)h(mo)s(di\014cation")d(of)i Fo(\037)2448 2504 y Fk(@)t(D)2549 2485 y Fj(6)2547 2520 y Fg(n)2594 2487 y Fq(\()p Fo(x)p Fq(\).)474 2607 y(W)-8 b(e)33 b(sho)m(w)h(that)1184 2819 y(lim)1110 2873 y Fk(n)p Fl(\000)-12 b(!)p Fe(+)p Fl(1)1409 2702 y Ff(Z)1455 2891 y Fk(D)1515 2868 y Fj(6)p Fd(\003)1513 2913 y Fj(1)1602 2819 y Fo(\037)1663 2836 y Fk(@)t(D)1764 2817 y Fj(6)1762 2853 y Fg(n)1809 2819 y Fq(\()p Fo(x)p Fq(\))p Fo(d)29 b Fq(~)-78 b Fo(!)2052 2834 y Fk(n)2154 2819 y Fq(=)2285 2702 y Ff(Z)2331 2891 y Fk(D)2391 2868 y Fj(6)p Fd(\003)2389 2913 y Fj(1)2478 2819 y Fo(\037)2539 2836 y Fk(S)2586 2817 y Fj(3)2625 2819 y Fq(\()p Fo(x)p Fq(\))p Fo(:)534 b Fq(\(4.91\))328 3082 y(Reely)-8 b(,)33 b(for)f(eac)m(h)h Fo(n)g Fq(w)m(e)g(can)g(\014nd)g(suc)m(h)h Fo(m)f Fq(that)814 3318 y Fn(j)870 3201 y Ff(Z)969 3318 y Fo(\037)1030 3333 y Fk(@)t(D)1129 3341 y Fg(n)1176 3318 y Fo(d)c Fq(~)-78 b Fo(!)1288 3333 y Fk(n)1357 3318 y Fn(\000)1456 3201 y Ff(Z)1556 3318 y 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b Fq(\()p Fo(n;)17 b(\017)p Fq(\))33 b(,)g(that)f(for)g(all)f Fo(m)d Fn(\025)g Fo(N)10 b Fq(\()p Fo(m;)17 b(\017)p Fq(\),)1308 4160 y Fn(j)1363 4043 y Ff(Z)1463 4160 y Fo(\037)1524 4175 y Fk(@)t(D)1623 4183 y Fg(n)1669 4160 y Fo(d)30 b Fq(~)-79 b Fo(!)1781 4175 y Fk(n)1850 4160 y Fn(\000)1950 4043 y Ff(Z)2049 4160 y Fo(f)2108 4119 y Fk(m)2097 4184 y(n)2175 4160 y Fo(d)29 b Fq(~)-78 b Fo(!)2287 4175 y Fk(n)2361 4160 y Fn(j)27 b Fo(<)h(\017;)328 4401 y Fq(\(the)33 b(Leb)s(esgue)h(dominated)d(con)m(v)m(ergence)k (theorem\).)474 4522 y(Th)m(us,)f(according)e(to)h(\(.\))43 b(w)m(e)34 b(\014nally)d(obtained)565 4721 y(1)p 463 4765 253 4 v 463 4857 a Fn(jj)c Fo(r)j Fn(jj)676 4822 y Fe(4)676 4879 y(6)780 4788 y Fq(=)101 b(lim)912 4838 y Fk(n)p Fl(\000)-12 b(!1)1155 4671 y Ff(Z)1201 4860 y Fk(@)t(D)1302 4841 y Fj(6)1300 4877 y Fg(n)1443 4721 y Fo(d!)1555 4736 y Fk(n)1602 4721 y Fq(\()p Fo(y)t Fq(\))p 1374 4765 426 4 v 1374 4857 a Fn(jj)27 b Fo(r)d Fn(\000)f Fo(y)31 b Fn(jj)1760 4822 y Fe(4)1760 4879 y(6)1864 4788 y Fq(=)1995 4671 y Ff(Z)2041 4860 y Fk(S)2088 4841 y Fj(3)2245 4721 y Fo(d)8 b Fq(~)-57 b Fo(!)s Fq(\()p Fo(y)t Fq(\))p 2153 4765 V 2153 4857 a Fn(jj)27 b Fo(r)e Fn(\000)e Fo(y)30 b Fn(jj)2539 4822 y Fe(4)2539 4879 y(6)2588 4788 y Fq(;)45 b Fo(r)30 b Fn(2)e Fo(D)2912 4747 y Fe(6)2909 4813 y(1)2973 4788 y Fn(\000)23 b Fo(B)3152 4747 y Fe(6)3192 4788 y Fq(\(1)p Fo(=)p Fq(2\))p Fo(:)3317 4965 y Fq(\(4.92\))1898 5214 y(35)p eop %%Page: 36 36 36 35 bop 474 631 a Fq(In)33 b(the)g(sequel)h(w)m(e)f(write)g Fo(R)1522 646 y Fl(1)1629 631 y Fq(instead)g(of)39 b(~)-56 b Fo(!)s Fq(.)474 751 y(So,if)32 b(w)m(e)h(restrict)g(ourselv)m(es)h (in)e(\(\))g(for)g Fo(h)c Fn(2)33 b Fq(,)f(w)m(e)i(\014nally)d(get)383 1018 y(\001)t(\()p Fo(h)p Fq(\))600 977 y Fl(\000)p Fe(1)750 1018 y Fq(=)882 901 y Ff(Z)928 1089 y Fk(S)975 1071 y Fj(3)1106 951 y Fo(dR)q Fq(\()p Fo(x)p Fq(\))p 1040 995 391 4 v 1040 1086 a(\001\()p Fo(h)22 b Fn(\000)h Fo(x)p Fq(\))1496 1018 y Fo(h)k Fn(2)h Fo(R)1747 1033 y Fe(4)1787 1018 y Fq(\()p Fo(r)m(;)17 b(r)1957 977 y Fl(\000)p Fe(1)2051 1018 y Fq(\)\()p Fo(r)30 b Fn(2)e Fq(\(0)p Fo(;)17 b Fq(1\)\))p Fo(:)739 b Fq(\(4.93\))328 1291 y(Let's)37 b(consider)g(a)e Fr(branc)m(h)43 b(of)f(the)f(hamiltonian)f(square)2723 1219 y Fn(p)p 2806 1219 28 4 v 72 x(\001)c Fq(and)h(the)f(induced)328 1411 y(map)27 b(of)h(measure)g(spaces)i(:)1385 1340 y Fn(p)p 1468 1340 V 71 x Fq(:\()p Fo(S)1599 1375 y Fe(3)1638 1411 y Fo(;)17 b(R)q Fq(\))28 b Fn(\000)-17 b(!)28 b Fq(\()p Fo(S)2115 1375 y Fe(3)2154 1411 y Fo(;)2198 1339 y Fn(p)p 2281 1339 V 72 x(\001)2308 1357 y Fl(\003)2348 1411 y Fo(R)q Fq(\).)42 b(Putting)27 b(in)g(\(.\))42 b Fo(h)3194 1375 y Fe(2)3262 1411 y Fq(instead)328 1532 y(of)32 b Fo(h)h Fq(w)m(e)g(\014nally)f(get:)1008 1798 y(\001)t(\()p Fo(h)1187 1757 y Fe(2)1227 1798 y Fq(\))55 b(=)1451 1681 y Ff(Z)1497 1869 y Fk(S)1544 1850 y Fj(3)1609 1730 y Fo(d)p Fq(\()1698 1658 y Fn(p)p 1781 1658 V 72 x(\001)p Fo(R)1883 1745 y Fl(1)1958 1730 y Fq(\)\()p Fo(y)t Fq(\))p 1609 1775 514 4 v 1631 1866 a(\001)t(\()p Fo(h)1810 1837 y Fe(2)1872 1866 y Fn(\000)23 b Fo(y)2024 1837 y Fe(2)2063 1866 y Fq(\))2188 1798 y(=)55 b Fn(H)t Fq(\()2445 1721 y Fn(p)p 2528 1721 28 4 v 77 x(\001)p Fo(R)2630 1813 y Fl(1)2705 1798 y Fq(\)\()p Fo(h)2837 1757 y Fe(2)2876 1798 y Fq(\))p Fo(;)376 b Fq(\(4.94\))328 2070 y(where)34 b Fo(h)28 b Fn(2)g Fo(R)862 2085 y Fk(H)929 2070 y Fq(\()p Fo(r)1014 2033 y Fl(\000)p Fe(1)1108 2070 y Fo(;)17 b(r)s Fq(\).)328 2288 y Fr(Theorem)37 b(2)49 b Fb(\()p Fr(On)37 b(the)h(existence)f(of)g(\(RH\)-\014xed)g(p)s(oin)m (ts)g(of)h Fn(F)10 b Fr(\))328 2409 y(Let)30 b Fo(A)e Fq(=)g Fo(A)p Fq(\()p Fo(x)p Fq(\))j Fr(b)s(e)f(a)h(p)s(ositiv)m(e)e (CID-amplitude)g(with)h Fo(A)p Fq(\(0\))d(=)h(0)i Fr(and)h Fo(A)p Fq(\()p Fo(x)p Fq(\))d Fn(&)328 2529 y Fq(0)41 b Fr(for)h Fn(j)33 b Fo(x)h Fn(j\025)g Fq(1)p Fr(.)62 b(Then)42 b(,)h(there)e(exists)g(suc)m(h)h(a)g(\014xed)g(p)s(oin)m(t)f Fo(!)k Fr(of)d Fn(F)51 b Fr(\(from)328 2650 y Fn(S)7 b Fq(\()p Fm(R)i Fq(\))p Fr(\))37 b(-)h(the)f(\(RH\)-\014xed)g(p)s(oin) m(t,)g(that)1316 2861 y Fq(\()p Fo(!)53 b Fn(\000)d Fo(G)p Fq(\)\()p Fo(x)p Fq(\))56 b(=)f Fo(A)p Fq(\()p Fo(x)p Fq(\))28 b(;)45 b Fo(x)28 b Fn(2)g Fm(R)2575 2876 y Fe(+)2634 2861 y Fo(:)656 b Fq(\(4.95\))328 3073 y Fr(In)37 b(particular)1569 3193 y Fo(!)t Fq(\(0\))54 b(=)h Fo(G)p Fq(\(0\))g(=)g(1)p Fo(:)909 b Fq(\(4.96\))328 3412 y Fr(Pro)s(of)p Fq(.)81 b(W)-8 b(e)45 b(start)h(from)e(the)h(recalling)e(of)i(the)g(fact,)j (that)d(w)m(e)h(ha)m(v)m(e)h(got)e(to)f(the)328 3532 y(desp)s(osition?)f(the)33 b(whole)f(family)1401 3744 y Fo(F)14 b Fq(\()p Fo(g)1563 3759 y Fk(A)1619 3744 y Fq(\)\()p Fo(x)p Fq(\))56 b(:=)f Fn(f)p Fo(f)2100 3759 y Fk(\025)2145 3744 y Fq(\()p Fo(x)p Fq(\))28 b(:)p Fn(j)f Fo(\025)h Fn(jg)768 b Fq(\(4.97\))328 3956 y(of)32 b(solutions)g(of)g (the)h(family)d(of)i(the)h(F)-8 b(o)m(x)32 b(equations:)1050 4167 y Fo(f)1098 4182 y Fk(\025)1143 4167 y Fq(\()p Fo(x)p Fq(\))50 b(+)g Fo(\025)1528 4141 y Fq(^)1507 4167 y Fo(f)1555 4182 y Fk(\025)1600 4167 y Fq(\()p Fo(x)p Fq(\))56 b(=)f Fo(g)1965 4182 y Fk(A)2022 4167 y Fq(\()p Fo(x)p Fq(\))28 b(:=)g Fo(G)p Fq(\()p Fo(x)p Fq(\))22 b(+)g Fo(A)p Fq(\()p Fo(x)p Fq(\))p Fo(:)446 b Fq(\(4.98\))474 4379 y(Since)30 b(the)g(consideration)e(of)h(the)h(pair)e(\()p Fm(R)8 b Fo(;)17 b(theLebesg)t(uemeasur)s(e)p Fq(\))30 b(seems)h(to)e(b)s(e) 328 4499 y Fr(unsu\016cien)m(t)p Fq(,)k(then)g(w)m(e)h(w)m(ork)f(in)f (this)g(momen)m(t)g(with)g(an)g(arbitrary)g(lo)s(cal)e(\014eld)j Fo(K)7 b Fq(.)474 4620 y(W)-8 b(e)36 b(mak)m(e)g(in)f(\(.\))53 b(the)37 b(substitution)e(:)50 b Fo(\025)33 b Fq(=)g(\001)2251 4583 y Fe(2)2251 4644 y Fk(K)2320 4620 y Fq(\()p Fo(k)s Fq(\);)17 b Fo(k)36 b Fn(2)d Fo(K)43 b Fq(,)37 b(with)e(\001)3176 4635 y Fk(K)3245 4620 y Fq(\()p Fo(k)s Fq(\))e Fn(6)p Fq(=)g(1)328 4740 y(to)f(obtain)1190 4823 y Fo(f)1238 4844 y Fe(\001)1297 4821 y Fj(2)1297 4868 y Fg(K)1355 4844 y Fe(\()p Fk(k)r Fe(\))1453 4823 y Fq(\()p Fo(x)p Fq(\))p 1190 4881 395 4 v 1247 4973 a(\001)1328 4939 y Fe(2)1328 4997 y Fk(K)1397 4973 y Fq(\()p Fo(k)s Fq(\))1644 4905 y(+)1791 4878 y(^)1770 4905 y Fo(f)1818 4925 y Fe(\001)1877 4902 y Fj(2)1877 4950 y Fg(K)1939 4905 y Fq(\()p Fo(k)s Fq(\)\()p Fo(x)p Fq(\))56 b(=)2420 4837 y Fo(g)2467 4852 y Fk(A)2523 4837 y Fq(\()p Fo(x)p Fq(\))p 2397 4881 280 4 v 2397 4973 a(\001)2478 4939 y Fe(2)2478 4997 y Fk(K)2547 4973 y Fq(\()p Fo(k)s Fq(\))2687 4905 y Fo(;)603 b Fq(\(4.99\))1898 5214 y(36)p eop %%Page: 37 37 37 36 bop 328 631 a Fq(where)39 b Fo(x)e Fn(2)g Fm(R)46 b Fq(and)38 b Fo(k)h Fn(2)e Fo(K)7 b Fq(.)59 b(In)m(tegrating)38 b(the)g(b)s(oth)f(sides)i(of)e(\(.\))59 b(with)37 b(resp)s(ect)i(to)328 751 y(the)g(left)f Fr(Haar)44 b(measure)39 b Fo(H)1487 766 y Fk(K)1594 751 y Fq(of)f(\()p Fo(K)r(;)17 b Fq(+\))37 b(and)i(applying)e(the)i(form)m(ula)e(\(4.74\))h(w)m(e)328 872 y(obtain)328 964 y Ff(Z)428 1081 y Fo(X)509 1096 y Fk(N)576 1081 y Fq(\()p Fo(K)7 b Fq(\)\001)823 1040 y Fl(\000)p Fe(2)823 1106 y Fk(K)917 1081 y Fq(\()p Fo(k)s Fq(\))p Fo(f)1095 1102 y Fe(\001)1154 1079 y Fj(2)1154 1126 y Fg(K)1213 1102 y Fe(\()p Fk(k)r Fe(\))1310 1081 y Fq(\()p Fo(x)p Fq(\))p Fo(dH)1573 1096 y Fk(K)1642 1081 y Fq(\()p Fo(k)s Fq(\))27 b(+)1903 964 y Ff(Z)2002 1081 y Fo(X)2083 1096 y Fk(N)2151 1081 y Fq(\()p Fo(K)7 b Fq(\)\001)2398 1040 y Fl(\000)p Fe(2)2398 1106 y Fk(K)2492 1081 y Fq(\()p Fo(k)s Fq(\))2643 1055 y(^)2622 1081 y Fo(f)2670 1109 y Fe(\001)2729 1079 y Fd(\000)p Fj(2)2729 1133 y Fg(K)2812 1109 y Fe(\()p Fk(k)r Fe(\))2909 1081 y Fq(\()p Fo(x)p Fq(\))p Fo(dH)3172 1096 y Fk(K)3240 1081 y Fq(\()p Fo(k)s Fq(\))56 b(=)f Fo(g)3604 1096 y Fk(A)3661 1081 y Fq(\()p Fo(x)p Fq(\))3809 964 y Ff(Z)3855 1153 y Fk(X)3913 1164 y Fg(N)3970 1153 y Fe(\()p Fk(K)5 b Fe(\))4110 1081 y Fq(\001)4191 1040 y Fl(\000)p Fe(2)4191 1106 y Fk(K)4286 1081 y Fq(\()p Fo(k)s Fq(\))p Fo(dH)4548 1096 y Fk(K)4616 1081 y Fq(\()p Fo(k)s Fq(\))27 b(=:)h Fo(M)4998 1096 y Fl(\000)p Fe(2)5093 1081 y Fq(\()p Fo(N)5 b(;)17 b(K)7 b Fq(\)\))p Fo(:)3268 1252 y Fq(\(4.100\))328 1373 y(In)30 b(the)f(sequel)i(it)d(is)h(con)m(v)m(enien)m(t)i(to)e (call)f Fo(M)1953 1388 y Fl(\000)p Fe(2)2047 1373 y Fq(\()p Fo(N)5 b(;)17 b(K)7 b Fq(\))29 b(the)h Fn(\000)p Fq(2)16 b Fn(\000)g Fo(K)7 b Fr(-momen)m(t)28 b Fq(of)h(the)328 1493 y(Haar)j(measure)h Fo(H)1030 1508 y Fk(K)1131 1493 y Fq(on)f(the)h Fo(X)1515 1508 y Fk(N)1582 1493 y Fq(\()p Fo(K)7 b Fq(\).)474 1614 y(No)m(w,)30 b(it)d(is)h(the)h(time)d(to)i (use)h Fr(compact-)j Fo(K)40 b Fr(-Hilb)s(ert)30 b(transform)e Fn(H)3186 1629 y Fk(K)3283 1614 y Fq(\(in)f(the)328 1734 y(case)33 b(of)g Fo(K)h Fq(=\))f(to)f(the)h(expressions:)1440 1903 y Ff(Z)1486 2091 y Fk(X)1544 2102 y Fg(N)1602 2091 y Fe(\()p Fk(K)5 b Fe(\))1752 1931 y Fo(f)1800 1959 y Fe(\001)1859 1929 y Fd(\006)p Fj(2)1859 1983 y Fg(K)1941 1959 y Fe(\()p Fk(k)r Fe(\))2039 1931 y Fq(\()p Fo(x)p Fq(\))p Fo(dH)2302 1946 y Fk(K)2370 1931 y Fq(\()p Fo(k)s Fq(\))p 1752 1997 749 4 v 1986 2088 a(\001)2067 2054 y Fe(2)2067 2112 y Fk(K)2136 2088 y Fq(\()p Fo(k)s Fq(\))2510 2020 y Fo(:)731 b Fq(\(4.101\))328 2280 y(According)32 b(to)g(the)h(Prop)s(osition)e(6.1?)43 b(w)m(e)34 b(ha)m(v)m(e)1112 2471 y(1)p 1008 2516 257 4 v 1008 2607 a(\001)t(\()p Fo(h)1187 2578 y Fe(2)1227 2607 y Fq(\))1330 2539 y(=)1462 2422 y Ff(Z)1508 2610 y Fk(S)1555 2591 y Fj(3)1740 2471 y Fo(dR)q Fq(\()p Fo(y)t Fq(\))p 1620 2516 495 4 v 1620 2607 a(\001)1701 2622 y Fk(H)1768 2607 y Fq(\()p Fo(h)1862 2578 y Fe(2)1924 2607 y Fn(\000)22 b Fo(y)2075 2578 y Fe(2)2124 2539 y Fq(\))55 b(=)g Fn(H)2432 2554 y Fk(H)2500 2539 y Fq(\()p Fo(R)2612 2554 y Fl(1)2687 2539 y Fq(\)\()p Fo(h)2819 2498 y Fe(2)2858 2539 y Fq(\))p Fo(;)345 b Fq(\(4.102\))328 2804 y(if)31 b Fo(h)d Fn(2)g Fo(X)676 2819 y Fk(N)743 2804 y Fq(\()p Fo(H)8 b Fq(\).)474 2925 y(Applying)32 b(the)h Fr(F)-9 b(ubini)36 b(theorem)c Fq(w)m(e)i(obtain:)343 3088 y Ff(Z)389 3276 y Fk(X)447 3287 y Fg(N)504 3276 y Fe(\()p Fk(H)5 b Fe(\))653 3116 y Fo(f)701 3144 y Fe(\001)760 3114 y Fd(\006)p Fj(2)760 3168 y Fg(H)843 3144 y Fe(\()p Fk(h)p Fe(\))942 3116 y Fq(\()p Fo(x)p Fq(\))p Fo(dH)1205 3131 y Fk(H)1272 3116 y Fq(\()p Fo(h)p Fq(\))p 653 3182 752 4 v 869 3273 a(\001)950 3288 y Fk(H)1017 3273 y Fq(\()p Fo(h)1111 3245 y Fe(2)1151 3273 y Fq(\))1497 3205 y(=)1656 3088 y Ff(Z)1702 3276 y Fk(X)1760 3287 y Fg(N)1818 3276 y Fe(\()p Fk(H)g Fe(\))1957 3205 y Fo(f)2005 3233 y Fe(\001)2064 3203 y Fd(\006)p Fj(2)2064 3257 y Fg(H)2146 3233 y Fe(\()p Fk(h)p Fe(\)\()p Fk(x)p Fe(\))2341 3205 y Fq(\()2379 3088 y Ff(Z)2425 3276 y Fk(S)2472 3258 y Fj(3)2676 3138 y Fo(dR)q Fq(\()p Fo(y)t Fq(\))p 2537 3182 533 4 v 2537 3273 a(\001)2618 3288 y Fk(H)2685 3273 y Fq(\()p Fo(h)2779 3245 y Fe(2)2841 3273 y Fn(\000)22 b Fo(y)2992 3245 y Fe(2)3031 3273 y Fq(\))3079 3205 y Fo(dH)3211 3220 y Fk(H)3278 3205 y Fq(\()p Fo(h)p Fq(\)\))27 b(=)3268 3381 y(\(4.103\))1111 3578 y(=)1270 3461 y Ff(Z)1316 3649 y Fk(S)1363 3630 y Fj(3)1418 3578 y Fo(dR)q Fq(\()p Fo(y)t Fq(\)\()1710 3461 y Ff(Z)1755 3649 y Fk(X)1813 3660 y Fg(N)1871 3649 y Fe(\()p Fk(H)5 b Fe(\))2019 3489 y Fo(f)2067 3517 y Fe(\001)2126 3487 y Fd(\006)p Fj(2)2126 3541 y Fg(H)2209 3517 y Fe(\()p Fk(h)p Fe(\))2309 3489 y Fq(\()p Fo(x)p Fq(\))p Fo(dH)t Fq(\()p Fo(h)p Fq(\))p 2019 3554 689 4 v 2129 3646 a(\001)t(\()p Fo(h)2308 3617 y Fe(2)2370 3646 y Fn(\000)23 b Fo(y)2522 3617 y Fe(2)2560 3646 y Fq(\))2718 3578 y(\))p Fo(:)474 3798 y Fq(But,)29 b(the)f(second)h (inner)e(in)m(tegral)f(in)h(the)g(ab)s(o)m(v)m(e)i(iterated)e(in)m (tegral,)g(w)m(e)h(can)g(write)328 3918 y(in)k(the)h(form:)554 4073 y Ff(Z)600 4262 y Fk(X)658 4273 y Fg(N)716 4262 y Fe(\(\))801 4091 y Fo(f)849 4118 y Fe(\001)908 4088 y Fd(\006)p Fj(2)908 4143 y Fg(H)991 4118 y Fe(\()1028 4091 y Fg(h)p 1028 4103 37 4 v 1029 4144 a(y)1074 4118 y Fe(\))1106 4091 y Fq(\()p Fo(x)p Fq(\))p Fo(dH)1369 4106 y Fk(H)1436 4091 y Fq(\()p Fo(h)p Fq(\))p 801 4167 767 4 v 833 4263 a(\001)914 4234 y Fe(2)954 4263 y Fq(\()p Fo(y)t Fq(\)\001)1163 4278 y Fk(H)1229 4263 y Fq(\(1)22 b Fn(\000)1448 4223 y Fk(h)p 1448 4239 41 4 v 1450 4297 a(y)1498 4263 y Fq(\))1605 4190 y(=)1709 4073 y Ff(Z)1755 4262 y Fk(X)1813 4273 y Fg(N)1871 4262 y Fe(\()p Fk(H)5 b Fe(\))2010 4190 y Fo(f)2058 4218 y Fe(\001)2117 4188 y Fd(\006)p Fj(2)2117 4242 y Fg(H)2199 4218 y Fe(\()p Fk(h)p Fe(\)\()p Fk(x)p Fe(\))p Fk(dH)t Fe(\()p Fk(h)p Fe(\))2587 4190 y Fq(\001)t(\(1)23 b Fn(\000)f Fo(h)2937 4149 y Fe(2)2977 4190 y Fq(\))p Fo(;)226 b Fq(\(4.104\))328 4482 y(according)32 b(to)g(the)h(form)m(ula)e(\(W\).)474 4612 y(But)749 4565 y Fk(dH)t Fe(\()p Fk(h)p Fe(\))p 678 4588 337 4 v 678 4646 a(\001)737 4657 y Fg(H)794 4646 y Fe(\(1)p Fl(\000)p Fk(h)952 4627 y Fj(2)987 4646 y Fe(\))1052 4612 y Fq(=:)d Fo(\026)1242 4627 y Fk(B)s(K)1399 4612 y Fq(is)k(a)h Fr(Bogolub)s(o\013-Krilo\013)i(measure)e Fq(of)f Fo(I)3087 4608 y Fd(\003)3128 4612 y Fq(,)g(i.e.)356 4788 y Ff(Z)402 4976 y Fk(X)460 4987 y Fg(N)517 4976 y Fe(\()p Fk(H)5 b Fe(\))666 4835 y Fo(f)714 4852 y Fe(\001)773 4833 y Fj(2)808 4852 y Fe(\()p Fk(h)p Fe(\))907 4835 y Fq(\()p Fo(x)p Fq(\))p Fo(dH)t Fq(\()p Fo(h)p Fq(\))p 666 4881 641 4 v 741 4973 a(\001)822 4988 y Fk(H)890 4973 y Fq(\(1)21 b Fn(\000)i Fo(h)1154 4944 y Fe(2)1194 4973 y Fq(\))1372 4905 y(=)1503 4788 y Ff(Z)1549 4976 y Fk(X)1607 4987 y Fg(N)1665 4976 y Fe(\()p Fk(H)5 b Fe(\))1803 4905 y Fo(f)1851 4921 y Fe(\001)1910 4902 y Fd(\000)p Fj(2)1993 4921 y Fe(\()p Fk(x)p Fe(\))p Fk(dH)t Fe(\()p Fk(h)p Fe(\))2286 4905 y Fq(\001)2367 4920 y Fk(H)2434 4905 y Fq(\(1)22 b Fn(\000)h Fo(h)2699 4864 y Fe(2)2738 4905 y Fq(\))p Fo(;)465 b Fq(\(4.105\))1898 5214 y(37)p eop %%Page: 38 38 38 37 bop 328 631 a Fq(since,)33 b(for)f(eac)m(h)i(in)m(tegrable)d (function)h Fo(\036)g Fq(the)h(follo)m(wing)d(calculus)i(is)g(true)684 741 y Ff(Z)730 930 y Fk(X)788 941 y Fg(N)846 930 y Fe(\()p Fk(H)5 b Fe(\))984 858 y Fo(\036)p Fq(\()p Fo(I)1123 873 y Fk(H)1186 854 y Fd(\003)1226 858 y Fq(\()p Fo(h)p Fq(\)\))p Fo(d\026)1506 873 y Fk(B)s(K)1631 858 y Fq(\()p Fo(h)p Fq(\))55 b(=)1949 741 y Ff(Z)1995 930 y Fk(X)2053 941 y Fg(N)2111 930 y Fe(\()p Fk(H)5 b Fe(\))2250 858 y Fo(\036)p Fq(\()p Fo(h)p Fq(\))p Fo(d)p Fq(\()p Fo(I)2580 817 y Fl(\003)2572 883 y Fk(H)2635 864 y Fd(\003)2674 858 y Fo(\026)2733 873 y Fk(B)s(K)2858 858 y Fq(\)\()p Fo(dh)p Fq(\))55 b(=)1485 1154 y(=)1616 1037 y Ff(Z)1662 1226 y Fk(X)1720 1237 y Fg(N)1778 1226 y Fe(\(\))1854 1154 y Fo(\036)p Fq(\()p Fo(h)p Fq(\))p Fo(d\026)2154 1169 y Fk(B)s(K)2278 1154 y Fq(\()p Fo(h)p Fq(\))p Fo(:)474 1367 y Fq(No)m(w,)34 b(observ)m(e)g(that)e Fo(f)1330 1382 y Fe(1)1402 1367 y Fq(w)m(asn't)i(de\014ned)g(y)m(et.)45 b(Therefore)33 b(w)m(e)h(de\014ne)1110 1621 y Fo(f)1158 1636 y Fe(1)1198 1621 y Fq(\()p Fo(x)p Fq(\))55 b(:=)h Fo(R)q Fq(\()p Fo(S)1722 1580 y Fe(3)1761 1621 y Fq(\))1816 1504 y Ff(Z)1862 1693 y Fk(X)1920 1704 y Fg(N)1977 1693 y Fe(\()p Fk(H)5 b Fe(\))2126 1551 y Fo(f)2174 1568 y Fe(\001)2233 1549 y Fj(2)2268 1568 y Fe(\()p Fk(h)p Fe(\))2367 1551 y Fq(\()p Fo(x)p Fq(\))p Fo(dH)2630 1566 y Fk(H)2697 1551 y Fq(\()p Fo(h)p Fq(\))p 2126 1598 704 4 v 2264 1690 a(\001)t(\(1)22 b Fn(\000)h Fo(h)2614 1661 y Fe(2)2654 1690 y Fq(\))2839 1621 y Fo(:)402 b Fq(\(4.106\))474 1874 y(Applying)36 b(the)i Fr(F)-9 b(ubini)41 b(theorem)36 b Fq(w)m(e)i(\014nally)e(get,)i(that)f(there)g Fr(exists)42 b(a)h(sin-)328 1995 y(gular)37 b(solution)31 b Fo(f)1082 2010 y Fe(1)1122 1995 y Fq(\()p Fo(x)p Fq(\))i(of)f(the)h(F)-8 b(o)m(x)32 b(equation)h(\(.\))43 b(:)1289 2198 y Fo(f)1337 2213 y Fe(1)1377 2198 y Fq(\()p Fo(x)p Fq(\))78 b(+)1758 2171 y(^)1739 2198 y Fo(f)1787 2213 y Fe(1)1827 2198 y Fq(\()p Fo(x)p Fq(\))83 b(=)55 b Fo(I)8 b Fq(\()p Fn(\000)p Fq(2\))p Fo(g)2472 2213 y Fk(A)2529 2198 y Fq(\()p Fo(x)p Fq(\))p Fo(:)581 b Fq(\(4.107\))328 2401 y(Finally)30 b(putting)1449 2521 y Fo(!)t Fq(\()p Fo(x)p Fq(\))55 b(:=)28 b Fo(f)1879 2536 y Fe(1)1919 2521 y Fq(\()p Fo(x)p Fq(\))50 b(+)2245 2495 y(^)2225 2521 y Fo(f)2273 2536 y Fe(1)2313 2521 y Fq(\()p Fo(x)p Fq(\))824 b(\(4.108\))328 2688 y(w)m(e)34 b(obtain)d(the)i(required)g(\(RH\)-\014xed)g(p)s(oin)m (t.)328 2897 y Fr(Prop)s(osition)i(3)49 b Fb(\()p Fr(On)38 b(the)f(p)s(ositivit)m(y)e(of)j(the)f(\(Rhfe\)-trace)p Fb(\).)328 3017 y(F)-7 b(or)33 b(e)-5 b(ach)34 b(\(RH\)-\014xe)-5 b(d)33 b(p)-5 b(oint)34 b Fo(!)1516 3032 y Fk(A)1607 3017 y Fb(of)g Fn(F)44 b Fb(asso)-5 b(ciate)g(d)33 b(with)h(a)g (CID-amplitude)e Fo(A)i Fb(as)g(in)328 3138 y(the)f(Th.2)g(,for)g(e)-5 b(ach)32 b Fo(a)c(>)g Fq(0)33 b Fb(and)f(c)-5 b(omplex)32 b Fo(s)c Fq(=)f Fo(u)18 b Fq(+)h Fo(iv)37 b Fb(with)c Fo(u)27 b Fn(2)h Fq([0)p Fo(;)17 b Fq(1)p Fo(=)p Fq(2])33 b Fb(and)f Fo(v)g(>)27 b Fq(0)p Fb(,)328 3258 y(the)35 b(fol)5 b(lowing)33 b(ine)-5 b(quality)35 b(holds:)504 3499 y Fo(T)14 b(r)619 3514 y Fk(an)703 3499 y Fq(\()p Fo(\020)8 b(;)17 b(A)p Fq(\)\()p Fo(s)p Fq(\))26 b(:=)1226 3382 y Ff(Z)1309 3408 y Fe(+)p Fl(1)1272 3571 y Fe(1)1438 3499 y Fq(\()p Fo(x)1531 3458 y Fk(u)p Fl(\000)p Fe(1)1689 3499 y Fn(\000)d Fo(x)1844 3458 y Fl(\000)p Fk(u)1944 3499 y Fq(\))p Fo(\022)s Fq(\()p Fo(A)p Fq(\)\()p Fo(x)p Fq(\))p Fo(sin)p Fq(\(2)p Fo(\031)t(v)t(l)r(nx)p Fq(\))p Fo(dx)57 b Fq(=)203 b(\(4.109\))1006 3736 y(=)28 b Fn(F)1182 3751 y Fe(+)1240 3736 y Fq(\()p Fo(\022)s Fq(\()p Fo(A)p Fq(\()p Fo(exp)p Fq(\)\)\))23 b Fn(\001)f Fo(E)6 b(xp)p Fq(\()p Fo(u)p Fq(\))22 b Fn(\001)g Fo(sin)p Fq(\(2)p Fo(\031)t(L)2546 3751 y Fk(v)2587 3736 y Fq(\))55 b Fo(>)h Fq(0)p Fo(;)328 3903 y Fb(,wher)-5 b(e)34 b(by)h Fo(E)6 b(xp)p Fq(\()p Fo(u)p Fq(\))35 b Fb(we)f(denote)-5 b(d)34 b(the)h(function)g Fo(e)2200 3867 y Fk(ux)2307 3903 y Fn(\000)23 b Fo(e)2452 3867 y Fe(\(1)p Fl(\000)p Fk(u)p Fe(\))p Fk(x)2716 3903 y Fb(of)35 b(the)g(variable)f Fo(x)p Fb(.)328 4112 y Fr(Pro)s(of)p Fq(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)475 4347 y Fo(T)14 b(r)590 4362 y Fk(an)674 4347 y Fq(\()p Fo(\020)8 b(;)17 b(A)p Fq(\)\()p Fo(s)p Fq(\))54 b(=)1257 4240 y Fl(1)1232 4264 y Ff(X)1226 4446 y Fk(n)p Fe(=1)1375 4230 y Ff(Z)1458 4257 y Fe(+)p Fl(1)1421 4419 y Fe(1)1588 4347 y Fq(\()p Fo(x)1681 4306 y Fk(u)p Fl(\000)p Fe(1)1839 4347 y Fn(\000)22 b Fo(x)1993 4306 y Fl(\000)p Fk(u)2093 4347 y Fq(\))p Fo(A)p Fq(\()p Fo(nx)p Fq(\))p Fo(sin)p Fq(\(2)p Fo(\031)t(v)t(l)r(nx)p Fq(\))p Fo(dx)30 b Fq(=)147 b(\(4.110\))328 4619 y(\(after)32 b(the)h(substitution)f Fo(x)c Fq(=)g Fo(e)1539 4583 y Fk(r)1577 4619 y Fq(\))976 4891 y(=)1083 4784 y Fe(+)p Fl(1)1086 4808 y Ff(X)1079 4990 y Fk(n)p Fe(=1)1233 4784 y(+)p Fl(1)1236 4808 y Ff(X)1229 4990 y Fk(n)p Fe(=1)1362 4891 y Fq(\()p Fo(e)1445 4850 y Fk(r)r(u)1546 4891 y Fn(\000)23 b Fo(e)1691 4850 y Fk(r)r Fe(\(1)p Fl(\000)p Fk(u)p Fe(\))1915 4891 y Fq(\))p Fo(A)p Fq(\()p Fo(ne)2167 4850 y Fk(r)2205 4891 y Fq(\))p Fo(sin)p Fq(\(2)p Fo(\031)t(v)t(r)s Fq(\))p Fo(dr)57 b Fq(=)1898 5214 y(38)p eop %%Page: 39 39 39 38 bop 1052 688 a Fq(:=)1186 580 y Fe(+)p Fl(1)1189 605 y Ff(X)1182 786 y Fk(n)p Fe(=1)1332 688 y Fn(F)1404 703 y Fe(+)1463 688 y Fq(\()p Fo(E)6 b(X)i(p)p Fq(\()p Fo(u)p Fq(\))21 b Fn(\001)h Fo(A)p Fq(\()p Fo(nexp)p Fq(\))h Fn(\001)f Fo(sin)p Fq(\(2)p Fo(\031)t(L)2698 703 y Fk(v)2739 688 y Fq(\)\))p Fo(:)328 942 y Fq(Consider)36 b(the)h Fr(amplitudes)e Fn(A)1562 957 y Fk(n)1608 942 y Fq(\()p Fo(r)s Fq(\))f(:=)f Fo(e)1946 906 y Fk(r)r(u)2025 942 y Fq(\(1)24 b Fn(\000)h Fo(e)2283 906 y Fk(r)r Fe(\(1)p Fl(\000)p Fe(2)p Fk(u)p Fe(\))2543 942 y Fq(\))f Fn(\001)g Fo(A)p Fq(\()p Fo(ne)2871 906 y Fk(r)2910 942 y Fq(\).)54 b(Since)3314 903 y Fk(d)p 3297 919 71 4 v 3297 977 a(dr)3377 942 y Fq(\(1)24 b Fn(\000)328 1063 y Fo(e)373 1027 y Fk(r)r Fe(\(1)p Fl(\000)p Fe(2)p Fk(u)p Fe(\))632 1063 y Fq(\))k(=)f(\(2)p Fo(u)11 b Fn(\000)g Fq(1\))p Fo(e)1175 1027 y Fk(r)r Fe(\(1)p Fl(\000)p Fe(2)p Fk(u)p Fe(\))1463 1063 y Fo(<)27 b Fq(0)g(for)g Fo(u)g Fn(2)h Fq(\(1)p Fo(=)p Fq(2)p Fo(;)17 b Fq(1],)28 b(then)g Fn(A)2620 1078 y Fk(n)2666 1063 y Fq(\()p Fo(r)s Fq(\))f(is)g(strictly)f(p)s (ositiv)m(e)328 1183 y(CID-amplitude)k(\(with)i Fn(A)1341 1198 y Fk(n)1388 1183 y Fq(\(0\))27 b(=)h(0\))328 1516 y Fp(5)161 b(The)53 b(Pro)t(of)i(of)e(the)g(Riemann)h(Hyp)t(othesis)328 1735 y Fr(Theorem)37 b(3)49 b Fb(Ther)-5 b(e)25 b(exists)h(such)g (functions)f Fo(F)2127 1750 y Fe(1)2167 1735 y Fq(\()p Fo(s)p Fq(\))h Fb(and)f Fo(F)2558 1750 y Fe(2)2598 1735 y Fq(\()p Fo(s)p Fq(\))h Fb(\()p Fq(\000)p Fo(\022)s(sinl)r(og)t Fr(-factors)p Fb(\))328 1855 y(and)i(r)-5 b(e)g(al)29 b(value)-5 b(d)28 b Fo(f)1029 1870 y Fe(1)1069 1855 y Fq(\()p Fo(s)p Fq(\))g Fb(and)g Fo(f)1450 1870 y Fe(2)1490 1855 y Fq(\()p Fo(s)p Fq(\))h Fb(\()p Fo(\022)j Fr(sinlog-factors)p Fb(\))c(with)h Fo(f)2727 1870 y Fe(1)2767 1855 y Fq(\()p Fo(s)p Fq(\))9 b Fn(\000)g Fo(f)3032 1870 y Fe(2)3071 1855 y Fq(\()p Fo(s)p Fq(\))27 b Fn(6)p Fq(=)h(0)g Fb(that)328 1976 y(the)g(fol)5 b(lowing)26 b Fr(Riemann)i(h)m(yp)s(othesis)h (functional)f(equation)g Fb(\(with)f(a)h Fr(rational)328 2096 y(term)36 b Fo(I)44 b Fb(and)37 b(the)f Fr(action)j(of)i Fo(Gal)r Fq(\()p Fm(C)24 b Fo(=)p Fm(R)8 b Fq(\))p Fb(\))37 b(\(R.h.f.e.)49 b(in)37 b(short\))f(holds)g(for)h 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b(v)t Fq(\))47 b(b)s(elongs)h(to)g(the)328 3872 y Fr(algebraic)38 b Fm(R)9 b Fr(-v)-6 b(ariete)33 b Fo(I)8 b Fq(\()p Fm(C)24 b Fq(\))33 b(.)48 b(The)35 b(main)d(determinan)m(t)i(of)f(\(3.38\))g (is)h Fo(I)8 b Fq(\()p Fo(s)p Fq(\))33 b(and)h(its)328 3992 y(solution)d(is)h(giv)m(en)h(b)m(y)542 4261 y Fo(p)591 4276 y Fe(1)686 4261 y Fq(=)55 b Fo(p)866 4276 y Fe(1)905 4261 y Fq(\()p Fo(I)8 b(m)p Fq(\()p Fo(s)p Fq(\);)17 b Fo(a)1296 4276 y Fe(1)1358 4261 y Fn(\000)22 b Fo(a)1508 4276 y Fe(2)1548 4261 y Fq(\))55 b(=)1782 4194 y(\()p Fo(a)1871 4209 y Fe(1)1933 4194 y Fn(\000)23 b Fo(a)2084 4209 y Fe(2)2123 4194 y Fq(\))p Fo(I)8 b Fq(\()p Fo(s)p Fq(\))p 1782 4238 552 4 v 1922 4330 a(3)p Fo(v)t(I)g Fq(\()p Fo(s)p Fq(\))2372 4261 y(=)2485 4194 y(\()p Fo(a)2574 4209 y Fe(1)2636 4194 y Fn(\000)22 b Fo(a)2786 4209 y Fe(2)2826 4194 y Fq(\))p 2485 4238 379 4 v 2625 4330 a(3)p Fo(v)2902 4261 y(>)27 b Fq(0)214 b(\(5.113\))328 4532 y(and)378 4782 y Fo(p)427 4797 y Fe(2)521 4782 y Fq(=)56 b Fo(p)702 4797 y Fe(2)741 4782 y Fq(\()p Fo(I)8 b(m)p Fq(\()p Fo(s)p Fq(\);)17 b Fo(a)1132 4797 y Fe(1)1186 4782 y Fn(\000)d Fo(a)1328 4797 y Fe(2)1368 4782 y Fq(\))55 b(=)1602 4714 y(\()p Fo(a)1691 4729 y Fe(2)1753 4714 y Fn(\000)23 b Fo(a)1904 4729 y Fe(1)1943 4714 y Fq(\))p Fo(T)14 b Fq(\()p Fo(s)p Fq(\))p 1602 4758 572 4 v 1752 4850 a(3)p Fo(v)t(I)8 b Fq(\()p Fo(s)p Fq(\))2211 4782 y(=)2325 4714 y(\()p Fo(a)2414 4729 y Fe(2)2476 4714 y Fn(\000)22 b Fo(a)2626 4729 y Fe(1)2666 4714 y Fq(\))p 2325 4758 379 4 v 2465 4850 a(3)p Fo(v)2741 4782 y Fq(=)28 b Fn(\000)p Fo(p)2971 4797 y Fe(1)3038 4782 y Fo(<)g Fq(0)p Fo(:)50 b Fq(\(5.114\))1898 5214 y(39)p eop %%Page: 40 40 40 39 bop 328 631 a Fq(Let's)46 b(denote)g(b)m(y)g Fo(J)1110 646 y Fe(1)1150 631 y Fq(\()p Fo(s)p Fq(\))f(the)g(in)m(tegral)f(in)h (the)g(righ)m(t-hand)g(side)g(of)g Fo(I)8 b(m)p Fq(\()p Fo(F)s(:p:t:f)d(:e)p Fq(\))328 751 y(for)37 b Fo(!)543 766 y Fk(A)637 751 y Fq(and)g(b)m(y)i Fo(J)1026 766 y Fe(2)1065 751 y Fq(\()p Fo(s)p 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b(=)f Fo(p)723 1495 y Fe(1)763 1480 y Fo(v)830 1362 y Ff(Z)913 1389 y Fl(1)876 1551 y Fe(1)988 1480 y Fq(\()p Fo(y)1078 1439 y Fk(p)1114 1448 y Fj(1)1147 1439 y Fk(v)r Fe(\()p Fk(u)p Fl(\000)p Fe(1\))1383 1480 y Fn(\000)9 b Fo(y)1521 1439 y Fl(\000)p Fk(p)1612 1448 y Fj(1)1646 1439 y Fk(v)r(u)1728 1480 y Fq(\))p Fo(sin)p Fq(\()p Fo(l)r(og)t(y)2122 1439 y Fk(p)2158 1448 y Fj(1)2191 1439 y Fk(v)2232 1480 y Fq(\))p Fo(\022)s Fq(\()p Fo(G)2433 1495 y Fe(1)2472 1480 y Fq(\)\()p Fo(y)2600 1439 y Fk(p)2636 1448 y Fj(1)2670 1439 y Fk(v)2710 1480 y Fq(\))p Fo(y)2800 1439 y Fe(\()p Fk(p)2863 1448 y Fj(1)2897 1439 y Fk(v)r Fl(\000)p Fe(1\))3056 1480 y Fo(dy)30 b Fq(=:)e Fo(J)3379 1439 y Fk(r)3370 1504 y Fe(1)3417 1480 y Fq(\()p Fo(s)p Fq(\))p Fo(:)3268 1634 y Fq(\(5.115\))328 1755 y(Analogously)-8 b(,)28 b(for)h Fo(p)1092 1770 y Fe(2)1159 1755 y Fo(<)f Fq(0)g(and)i Fo(v)h(>)d Fq(0)g(w)m(e)j(c)m(hange)f(v)-5 b(ariables)27 b(in)i Fo(J)2811 1770 y Fe(2)2879 1755 y Fq(according)g(to)f(the)328 1875 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Fq(\(cf.e.g.[Fi]\))g(w)m(e)h(obtain)f(that)h(there)g(exists)g (suc)m(h)i Fo(y)2831 4395 y Fk(i)2907 4380 y Fq(=)h Fo(y)3079 4395 y Fk(i)3107 4380 y Fq(\()p Fo(s;)17 b(a)p Fq(\))44 b(from)328 4500 y([1)p Fo(;)17 b Fq(+)p Fn(1)p Fq(\))31 b(that)348 4752 y Fo(J)402 4767 y Fk(i)430 4752 y Fq(\()p Fo(c)510 4767 y Fk(i)538 4752 y Fq(\()p Fo(s)p Fq(\)\))c(=)h Fo(p)878 4767 y Fk(i)906 4752 y Fo(v)t(sin)p Fq(\()p Fo(l)r(og)t(y)1313 4705 y Fk(p)1349 4715 y Fg(i)1374 4705 y Fk(v)1410 4682 y Fj(2)1309 4777 y Fk(G)1449 4752 y Fq(\))p Fo(\022)s Fq(\()p Fo(G)1650 4767 y Fk(i)1678 4752 y Fq(\)\()p Fo(y)1806 4705 y Fk(p)1842 4715 y Fg(i)1867 4705 y Fk(v)1802 4775 y(i)1908 4752 y Fq(\))p Fo(y)1998 4705 y Fk(p)2034 4715 y Fg(i)2059 4705 y Fk(v)r Fl(\000)p Fe(1)1994 4775 y Fk(i)2212 4752 y Fn(\001)2262 4635 y Ff(Z)2345 4661 y Fl(1)2308 4824 y Fe(1)2420 4752 y Fq(\()p Fo(y)2510 4711 y Fk(p)2546 4721 y Fg(i)2571 4711 y Fk(v)r Fe(\()p Fk(u)p Fl(\000)p Fe(1\))p Fl(\000)p Fk(a)2884 4721 y Fg(i)2938 4752 y Fn(\000)23 b Fo(y)3090 4711 y Fl(\000)p Fk(p)3181 4721 y Fg(i)3206 4711 y Fk(v)r(u)p Fl(\000)p Fk(a)3375 4721 y Fg(i)3406 4752 y Fq(\))p Fo(dy)3268 4907 y Fq(\(5.118\))1898 5214 y(40)p eop %%Page: 41 41 41 40 bop 328 631 a Fq(where)35 b(b)m(y)f Fo(c)789 646 y Fe(1)862 631 y Fq(w)m(e)g(denoted)g(the)g(iden)m(tit)m(y)f(map)g Fo(id)2207 659 y Fm(C)2310 631 y Fq(and)g(b)m(y)i Fo(c)2679 646 y Fe(2)2751 631 y Fq(the)f(complex)f(conju-)328 751 y(gation)h Fo(c)p Fq(.)51 b(\(Ma)m(y)35 b(b)s(e)h(ev)m(en)g(these)h (elemen)m(tary)e(analytic)f(facts)h(w)m(e)h(w)m(ould)f(not)g(recall)328 872 y(and)e(write)f(but)h(w)m(e)g(enclosed)h(them)e(for)g(the)h(sak)m (e)h(of)e(completness\).)474 992 y(Ob)m(viously)44 b(w)m(e)g(can)g(alw) m(a)m(ys)h(tak)m(e)f(arti\016cial)d Fo(a)2308 1007 y Fk(i)2380 992 y Fq(in)i(\(3.38\))g(in)g(suc)m(h)i(a)e(manner)328 1112 y(that)35 b(in)m(tegrals)f(in)g(the)i(righ)m(t-hand)e(side)h(of)f (\(3.44\))h(are)g Fr(con)m(v)m(ergen)m(t)p Fq(.)51 b(Let's)36 b(de\014ne)328 1233 y Fo(\022)s(sinl)r(og)t Fr(-factors)d Fq(as)490 1463 y Fo(f)538 1478 y Fk(i)567 1463 y Fq(\()p Fo(s)p Fq(\))27 b(=)h Fo(f)868 1478 y Fk(i)896 1463 y Fq(\()p Fo(I)8 b(m)p Fq(\()p Fo(s)p Fq(\)\))27 b(=)h Fo(p)1410 1478 y Fk(i)1438 1463 y Fo(I)8 b(m)p Fq(\()p Fo(s)p Fq(\))p Fo(sin)p Fq(\()p Fo(l)r(og)t(y)2052 1412 y Fk(p)2088 1422 y Fg(i)2113 1412 y Fk(I)d(m)2211 1388 y Fj(2)2246 1412 y Fe(\()p Fk(s)p Fe(\))2048 1486 y Fk(i)2337 1463 y Fq(\))p Fo(\022)s Fq(\()p Fo(G)2538 1478 y Fk(i)2567 1463 y Fq(\)\()p Fo(y)2695 1412 y Fk(p)2731 1422 y Fg(i)2755 1412 y Fk(I)g(m)p Fe(\()p Fk(s)p Fe(\))2691 1486 y Fk(i)2945 1463 y Fq(\))p Fo(y)3035 1412 y Fk(p)3071 1422 y Fg(i)3096 1412 y Fk(I)g(m)p Fe(\()p Fk(s)p Fe(\))p Fl(\000)p Fe(1)3031 1486 y Fk(i)3376 1463 y Fq(;)3268 1583 y(\(5.119\))328 1703 y Fo(i)46 b Fq(=)g(1)p Fo(;)17 b Fq(2.)74 b(If)43 b(no)m(w)h(w)m(e)g(write)f(do)m(wn)h(\(Fptfe\))f(\(2.30\))g(in)f(the)i (case)g(of)f(\014xed)h(p)s(oin)m(ts)328 1824 y Fo(!)31 b Fq(=)c Fo(G)600 1839 y Fk(i)661 1824 y Fq(,)33 b(c)m(hange)g(v)-5 b(ariables)31 b(as)i(ab)s(o)m(v)m(e)h(and)e(use)i(the)f(ab)s(o)m(v)m(e) g(factors)g(w)m(e)g(obtain)415 2089 y Fo(I)8 b(m)p Fq(\(\(\000\()p Fo(G)803 2104 y Fk(i)831 2089 y Fq(\))p Fo(\020)g Fq(\)\()p Fo(s)p Fq(\)\))26 b(=)1454 2022 y Fo(I)8 b Fq(\()p Fo(s)p Fq(\))p 1258 2066 565 4 v 1258 2158 a(2)27 b Fn(j)h Fo(s)p Fq(\()p Fo(s)22 b Fn(\000)g Fq(1\))28 b Fn(j)1784 2129 y Fe(2)1855 2089 y Fq(+)22 b Fo(f)2001 2104 y Fk(i)2029 2089 y Fq(\()p Fo(s)p Fq(\))2168 1972 y Ff(Z)2251 1998 y Fe(+)p Fl(1)2214 2161 y Fe(1)2380 2089 y Fq(\()p Fo(y)2470 2048 y Fk(p)2506 2058 y Fg(i)2531 2048 y Fk(v)r Fe(\()p Fk(u)p Fl(\000)p Fe(1\))p Fl(\000)p Fk(a)2844 2058 y Fg(i)2899 2089 y Fn(\000)g Fo(y)3050 2048 y Fk(p)3086 2058 y Fg(i)3111 2048 y Fk(v)r(u)p Fl(\000)p Fk(a)3280 2058 y Fg(i)3312 2089 y Fq(\))p Fo(dy)t(:)3268 2265 y Fq(\(5.120\))328 2386 y(Hence)34 b(w)m(e)f(obtain)f(the)h(follo)m(wing) d(system)j(of)f(t)m(w)m(o)i(eqations)1151 2651 y Fo(I)8 b(m)p Fq(\(\(\000\()p 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Fe(2)1519 3254 y Fq(\))p Fo(f)1605 3269 y Fe(1)1644 3254 y Fo(\020)g Fq(\)\()p Fo(c)p Fq(\()p Fo(s)p Fq(\)\)\))27 b(=)2204 3186 y Fn(\000)p Fo(f)2329 3201 y Fe(1)2369 3186 y Fq(\()p Fo(s)p Fq(\))p Fo(I)8 b Fq(\()p Fo(s)p Fq(\))p 2151 3230 V 2151 3322 a(2)28 b Fn(j)f Fo(s)p Fq(\()p Fo(s)22 b Fn(\000)h Fq(1\))k Fn(j)2677 3293 y Fe(2)2726 3254 y Fq(+)673 3538 y(+\()p Fo(f)835 3553 y Fe(1)874 3538 y Fo(f)922 3553 y Fe(2)962 3538 y Fq(\)\()p Fo(I)8 b(m)p Fq(\()p Fo(s)p Fq(\)\))1351 3421 y Ff(Z)1433 3447 y Fe(+)p Fl(1)1397 3610 y Fe(1)1563 3538 y Fq(\()p Fo(y)1653 3497 y Fk(p)1689 3506 y Fj(2)1722 3497 y Fk(I)d(m)p Fe(\()p Fk(s)p Fe(\)\()p Fk(Re)p Fe(\()p Fk(s)p Fe(\))p Fl(\000)p Fe(1\))p Fl(\000)p Fk(a)2316 3506 y Fj(2)2380 3538 y Fn(\000)23 b Fo(y)2532 3497 y Fk(p)2568 3506 y Fj(2)2601 3497 y Fk(I)5 b(m)p Fe(\()p Fk(s)p Fe(\))p Fk(Re)p Fe(\()p Fk(s)p Fe(\))p Fl(\000)p Fk(a)3051 3506 y Fj(2)3092 3538 y Fo(dy)t(:)328 3740 y Fq(Let,)33 b(s)g(in)m(tro)s(duce)f(the)h(\000)p 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b(ob)m(viously)f(in)m(tegrals)f(in)h(the)h (righ)m(t-hand)f(sides)h(\(3.44\))f(and)328 4533 y(\(3.39\))j(of)g Fo(J)775 4548 y Fk(i)835 4533 y Fq(are)h Fr(equal)g Fq(eac)m(h)g(other) g(then)932 4749 y Fo(f)980 4764 y Fe(1)1019 4749 y Fq(\()p Fo(I)8 b(m)p Fq(\()p Fo(s)p Fq(\)\))22 b Fn(\000)h Fo(f)1523 4764 y Fe(2)1562 4749 y Fq(\()p Fo(I)8 b(m)p Fq(\()p Fo(s)p Fq(\)\))56 b Fn(6)p Fq(=)27 b(0)55 b Fo(f)11 b(or)30 b(R)q(e)p Fq(\()p Fo(s)p Fq(\))e Fn(2)g Fq(\(0)p Fo(;)17 b Fq(1)p Fo(=)p Fq(2\))p Fo(;)328 4965 y Fq(whic)m(h)33 b(pro)m(v)m(es)h(Theorem)f(2.)1898 5214 y(41)p eop %%Page: 42 42 42 41 bop 328 631 a Fr(Remark)37 b(10)49 b Fb(The)26 b(fol)5 b(lowing)26 b(natur)-5 b(al)27 b(question)g(arises)f(:)41 b(why)27 b(the)g(imp)-5 b(ortant)27 b(is)g(the)328 751 y(imaginary)36 b(p)-5 b(art)37 b(of)g(t.f.e)g(but)h(the)f(r)-5 b(e)g(al)37 b(one)f(is)h(not)g(\()g(so)g(it)g(is)g(a)g(str)-5 b(ange)37 b Fr(violation)328 872 y(\(breaking\))32 b(of)i(symmetry)c Fb(in)h(the)h(R)n(iemann)d(hyp)-5 b(othesis)31 b(pr)-5 b(oblem\).)43 b(May)32 b(b)-5 b(e)31 b(the)328 992 y(explanation)41 b(is)g(the)h(fol)5 b(lowing)41 b(:)59 b(in)42 b(the)g(inte)-5 b(gr)g(al)41 b Fo(I)8 b(m)p Fq(\()p Fo(J)h Fq(\()p Fo(s;)17 b(!)t Fq(\)\))40 b(=)h Fo(I)8 b(m)p Fq(\()3133 921 y Ff(R)3172 1020 y Fm(R)3242 992 y Fq(\()p Fo(x)3335 956 y Fk(s)p Fl(\000)p Fe(1)3490 992 y Fq(+)328 1112 y Fo(x)383 1076 y Fl(\000)p Fk(s)475 1112 y Fq(\)\))p Fo(\022)s Fq(\()p Fo(!)t Fq(\)\()p Fo(x)p Fq(\))p Fo(dx)33 b Fb(the)g(exp)-5 b(onential)31 b(factor)i(has)f(the)h(form)g Fq(\()p Fo(x)2618 1076 y Fk(u)p Fl(\000)p Fe(1)2771 1112 y Fn(\000)19 b Fo(x)2922 1076 y Fl(\000)p Fk(u)3022 1112 y Fq(\))p Fb(,)33 b(wher)-5 b(e)g(as)32 b(in)328 1233 y Fo(R)q(e)p Fq(\()p Fo(J)9 b Fq(\))36 b Fb(the)g(exp)-5 b(onential)35 b(factor)h(is)g Fq(\()p Fo(x)1781 1197 y Fk(u)p Fl(\000)p Fe(1)1939 1233 y Fq(+)23 b Fo(x)2093 1197 y Fl(\000)p Fk(u)2193 1233 y Fq(\))p Fb(.)48 b(So,)36 b(in)g(the)g(imaginary)f(p)-5 b(art)36 b(ap-)328 1353 y(p)-5 b(e)g(ars)29 b(the)h(arithmetic)f(op)-5 b(er)g(ation)29 b Fq(")11 b Fn(\000)g Fq(")p Fb(,)31 b(wher)-5 b(e)g(as)29 b(in)g(the)h(r)-5 b(e)g(al)29 b(p)-5 b(art)30 b Fq(")11 b(+)g(")p Fb(.)42 b(Now,)31 b(let's)328 1474 y(observe)37 b(that)i(the)f(formula)g(:)52 b Fo(l)r Fq(\([)p Fo(a;)17 b(b)p Fq(]\))35 b(:=)f(\()p Fo(b)25 b Fn(\000)g Fo(a)p Fq(\))38 b Fb(gives)g(the)g(very)h(imp)-5 b(ortant)38 b(\(and)328 1594 y(fundamental\))29 b(notion)h(of)g(the)g Fr(Leb)s(esgue)k(measure)d Fb(on)e Fm(R)9 b Fb(,)31 b(wher)-5 b(e)g(as)30 b(the)g(formula)g(:)328 1724 y Fo(c)p Fq(\([)p Fo(a;)17 b(b)p Fq(]\))28 b(:=)805 1677 y Fe(\()p Fk(b)p Fe(+)p Fk(a)p Fe(\))p 805 1700 178 4 v 876 1758 a(2)1022 1724 y Fb(gives)h(less)g(imp)-5 b(ortant)30 b(notion)f(of)h(the)f Fr(cen)m(tre)h Fb(of)f(interval)h(.)43 b(Mor)-5 b(e-)328 1844 y(over)28 b(the)g(fade)g(wave)f(volume)h(of)g(the)g(form)g(:)42 b Fo(A)2089 1859 y Fk(s)2153 1844 y Fq(:=)2284 1773 y Ff(R)2339 1799 y Fe(+)p Fl(1)2323 1869 y Fe(0)2485 1844 y Fo(A)p Fq(\()p Fo(x)p Fq(\))p Fo(sin)p Fq(\()p Fo(ax)p Fq(\))p Fo(dx)30 b Fb(for)e(a)h(non-)328 1964 y(incr)-5 b(e)g(asing)35 b(function)h Fo(A)p Fq(\()p Fo(x)p Fq(\))h Fb(is)f(always)g Fr(non-v)-6 b(anishing)37 b Fb(,)g(wher)-5 b(e)g(as)35 b(non-vanishing)328 2085 y(of)c(the)h(inte)-5 b(gr)g(al)32 b Fo(A)1016 2100 y Fk(c)1078 2085 y Fq(:=)1209 2014 y Ff(R)1264 2040 y Fl(1)1248 2110 y Fe(0)1355 2085 y Fo(A)p Fq(\()p Fo(x)p Fq(\))p Fo(cos)p Fq(\()p Fo(ax)p Fq(\))h Fb(ne)-5 b(e)g(ds)31 b(some)g(additional)g(c)-5 b(onditions)30 b(on)i Fo(A)p Fb(.)328 2417 y Fp(References)377 2637 y Fq([1])48 b([AAR])36 b(Anon)m(ymous)f(referee)i(,)p Fb(R)-5 b(ep)g(ort)37 b(on)g(the)g(p)-5 b(ap)g(er)36 b(A)-5 b(cta)38 b(A)n(rithmetic)-5 b(a)37 b(3765)528 2757 y("A)30 b(pr)-5 b(o)g(of)29 b(of)g(the)g(R)n(iemann)f(hyp)-5 b(othesis")29 b(by)g(A.)h(M)r(\030)-47 b(adr)-5 b(e)g(cki)10 b Fq(,)27 b(Ma)m(y)h(7th)e(\(2000\),1-)528 2877 y(3.)377 3081 y([2])48 b([BC])32 b(Bost)f(R.)g(and)g(Connes)i(A.,)e Fb(He)-5 b(cke)33 b(algebr)-5 b(as,)33 b(typ)-5 b(e)34 b(III)e(factors)h(and)g(phase)528 3201 y(tr)-5 b(ansitions)37 b(with)g(sp)-5 b(ontane)g(ous)36 b(symmetry)h(br)-5 b(e)g(aking)36 b(in)h(numb)-5 b(er)36 b(the)-5 b(ory)37 b Fq(,)f(Se-)528 3321 y(lecta)c(Math.)h(1\(1995\),)f(411-457.)377 3525 y([3])48 b([BCKV])30 b(Bump)e(D.,)h(Choi,)f(Kurlb)s(erg)g(and)g(V)-8 b(aaler,)29 b Fb(A)i(lo)-5 b(c)g(al)30 b(R)n(iemann)g(hyp)-5 b(oth-)528 3645 y(esis)8 b Fq(,)33 b(Math.)g(Zeitsc)m(hrift)e (223\(2000\),1-19.)377 3849 y([4])48 b([BK])30 b(Bogolub)s(o\013)d (N.N.)j(and)f(Krilo\013)e(N.M.,)k Fb(L)-5 b(a)32 b(the)-5 b(orie)g(gener)g(ale)30 b(de)i(la)f(mesur)-5 b(e)528 3969 y(dans)35 b(son)g(applic)-5 b(ation)34 b(a)h(l'etude)g(des)g (systems)g(dynamiques)f(de)h(la)g(me)-5 b(c)g(anique)528 4089 y(non-line)g(ar)g(e)34 b Fq(,)e(Ann.)i(of)e (Math.38\(1937\),65-113.)377 4293 y([5])48 b([dB1])42 b(de)f(Branges)h(L.,)h Fb(Hilb)-5 b(ert)43 b(sp)-5 b(ac)g(es)42 b(of)h(entir)-5 b(e)42 b(functions)8 b Fq(,)43 b(Pren)m(tice-Hall,)528 4413 y(Englew)m(o)s(o)s(d)33 b(Cli\013s,)e(1968.)377 4617 y([6])48 b([dB2])25 b(de)h(Branges)f(L.,)i Fb(The)g(R)n(iemann)f (hyp)-5 b(othesis)27 b(for)h(Hilb)-5 b(ert)28 b(sp)-5 b(ac)g(es)27 b(of)g(entir)-5 b(e)528 4737 y(functions)35 b Fq(,)d(BAMS)i(15\(1986\),1-17.)1898 5214 y(42)p eop %%Page: 43 43 43 42 bop 377 631 a Fq([7])48 b([dB3])34 b(de)g(Branges)g(L.,)g Fb(The)h(c)-5 b(onver)g(genc)g(e)34 b(of)i(Euler)g(pr)-5 b(o)g(ducts)35 b Fq(,)f(J.)g(F)-8 b(unctional)528 751 y(Analysis)33 b(107\(1992\),)e(122-210.)377 946 y([8])48 b([dB4])27 b(de)h(Branges)f(L.,)h Fb(A)i(c)-5 b(onje)g(ctur)g(e)29 b(which)g(implies)f(the)h(R)n(iemann)f(hyp)-5 b(othesis)528 1066 y Fq(,)33 b(J.)g(F)-8 b(unctional)30 b(Analysis)j (121\(1994\),117-184.)377 1261 y([9])48 b([dB5])94 b(de)f(Branges)h (L.,)108 b Fb(A)91 b(pr)-5 b(o)g(of)90 b(of)g(the)h(R)n(iemann)e(hyp)-5 b(othesis)90 b Fq(,)528 1381 y(preprin)m(t\(1999\),1-31.)328 1576 y([10])48 b([Br])38 b(Brou)m(w)m(er)h(L.E.J.,)g Fb(De)g(onb)-5 b(etr)g(ouwb)g(aarheid)38 b(der)h(lo)-5 b(gische)38 b(princip)-5 b(es,)p Fq(,)38 b(Ti-)528 1696 y(jdsc)m(hrift)33 b(v)m(o)s(or)g(w)m(ejsb)s(egeerte)h (2\(1908\),152-158.)328 1890 y([11])48 b([Ch])31 b(Chandrasekharan)h (K.,)e Fr(Arithmetic)h(functions)p Fq(,)f(1970,)g(B-H-NY)f(\(Die)528 2011 y(Grundlehren)k(der)g(mathematisc)m(hen)f(Wissensc)m(haften,)j (Band)e(167\).)328 2205 y([12])48 b([CJA])41 b(Anon)m(ymous)g(referee,) i Fb(Cr)-5 b(el)5 b(le's)41 b(Journal)h(R)-5 b(ep)g(ort)42 b(on)f("A)h(pr)-5 b(o)g(of)41 b(of)h(the)528 2326 y(R)n(iemann)34 b(hyp)-5 b(othesis")34 b(by)h(A.)g(M)r(\030)-47 b(adr)-5 b(e)g(cki)10 b Fq(,F)-8 b(ebruary)33 b(26\(1999\),1-3.)328 2520 y([13])48 b([CL])31 b(Conrey)h(J.B.)f(and)g(Li)e(Xian-Jin,)g Fb(A)k(note)g(on)g(some)f(p)-5 b(ositivity)33 b(c)-5 b(onditions)528 2640 y(r)g(elate)g(d)35 b(to)g(zeta)g(and)f (L-functions)h Fq(,)d(preprin)m(t)h(\(2000\),1-9.)328 2835 y([14])48 b([De])39 b(Devlin)f(K.,)j Fb(Go)-5 b(o)g(dbye,)42 b(Desc)-5 b(artes,)41 b(The)f(End)h(of)f(L)-5 b(o)g(gic)41 b(and)f(the)h(Se)-5 b(ar)g(ch)528 2955 y(for)35 b(a)g(New)g(Cosmolo)-5 b(gy)33 b(of)i(the)g(Mind)10 b Fq(,)33 b(John)f(Wiley)g(and)h(Sons,)g (Inc.,)h(1997.)328 3150 y([15])48 b([DG])30 b(Dugundji)e(J.)j(and)f (Granas)f(A.,)i Fb(Fixe)-5 b(d)32 b(p)-5 b(oint)32 b(the)-5 b(ory)62 b Fq(,V)-8 b(ol.I,)30 b(Monogra\014e)528 3270 y(Matemat)m(yczne)k(61,)f(PWN,)g(W)-8 b(arsza)m(w)m(a)34 b(,)e(1982.)328 3465 y([16])48 b([Ed])34 b(Edw)m(ards)g(H.)f(M.,)g Fb(R)n(iemann)-10 b('s)33 b(zeta)i(function)g Fq(,)d(Academic)h(Press,) h(NY-L,)528 3585 y(1974.)328 3779 y([17])48 b([F)-8 b(a])35 b(F)-8 b(altings)32 b(G.,)j Fb(End)5 b(lichkeitssatze)35 b(fur)i(ab)-5 b(elsche)36 b(V)-7 b(arietaten)36 b(ub)-5 b(er)37 b(Zahlko-)528 3900 y(rp)-5 b(en)35 b Fq(,)d(In)m(v)m(en)m(t.)j (Math.)e(73)f(\(1983\),349-366;)e(Erratum:)43 b(75\(1984\),p.381.)328 4094 y([18])48 b([F)-8 b(e])31 b(F)-8 b(eller)29 b(W.,)j Fb(A)n(n)h(intr)-5 b(o)g(duction)33 b(to)g(pr)-5 b(ob)g(ability)33 b(the)-5 b(ory)33 b(and)g(its)g(applic)-5 b(ations)528 4215 y(II)16 b Fq(,)32 b(John)h(Wiley)f(and)g(Sons)h(Inc.,)h(NY,)f (1966.)328 4409 y([19])48 b([Fi])33 b(Fic)m(h)m(tenholtz)h(G.)f(M.,)h Fb(The)i(c)-5 b(ourse)35 b(of)h(di\013er)-5 b(ential)35 b(and)g(inte)-5 b(gr)g(al)36 b(c)-5 b(alculus)528 4529 y(II)34 b Fq(,)f(Fizmatgiz,)d(Mosk)m(ou-Leningrad)j(,1960.)328 4724 y([20])48 b([Go])35 b(Go)s(del)e(K.,)i Fb(Ub)-5 b(er)37 b(formal)f(unentscheidb)-5 b(ar)g(e)36 b(Satze)g(der)h (Principia)f(Math-)528 4844 y(ematic)-5 b(a)36 b(und)g(verwandter)f (Systeme.)h(I)g Fq(,)e(Monatshefte)h(fur)f(Mathematik)f(und)528 4965 y(Ph)m(ysic)h(38\(1931\),)d(173-198.)1898 5214 y(43)p eop %%Page: 44 44 44 43 bop 328 631 a Fq([21])48 b([H])28 b(Hartman)d(S.,)k Fb(Wstep)g(do)h(analizy)f(harmonicznej)14 b Fq(\(in)24 b(P)m(olish\),)j(PWN)h(1978?)328 834 y([22])48 b([KKM])59 b(Krasno)m(v)f(M.L.,Kisielev)g(A.I.)g(and)h(Mak)-5 b(arenk)m(o)59 b(G.I.,)p Fb(Inte)-5 b(gr)g(alnyje)528 955 y(ur)g(avnienia)p Fq(,)32 b(Nauk)-5 b(a,)33 b(Mosk)m(ou)h(1968.)328 1158 y([23])48 b([KV])34 b(Karatsuba)f(A.)h(A.)g(and)f(V)-8 b(oronin)33 b(S.)h(M.)f(,)h Fb(The)h(R)n(iemann)f(zeta)i(function)528 1279 y Fq(,)d(W)-8 b(alter)32 b(de)h(Gruy)m(er')g(Exp)s(ositions)f(in)g (Mathematics,)g(B-NY,)h(1992.)328 1482 y([24])48 b([ML])36 b(Martin-Lof)d(P)-8 b(.,)36 b Fb(Notes)h(on)g(c)-5 b(onstructive)36 b(mathematics)8 b Fq(,)35 b(Almquist)f(and)528 1602 y(Wiksell,)e(Sto)s (c)m(kholm)g(\(1970\).)328 1806 y([25])48 b([Ma])h(Maurin)e(K.,)52 b Fb(A)n(naliza)c(II)h(\(Wst\030)-45 b(ep)49 b(do)f(analizy)h(glob)-5 b(alnej\))46 b Fq(\(in)i(P)m(olish\))528 1926 y(\(Analysis)37 b(I)s(I)g(\(An)h(in)m(tro)s(duction)e(to)g(global)f(analysis\)\))i(,)h (Bibliotek)-5 b(a)35 b(Matem-)528 2046 y(at)m(yczna)f(\(BM\))f(41,)f (PWN)h(,)g(W)-8 b(arsza)m(w)m(a,1971.)328 2250 y([26])48 b([MaT])23 b(Madrec)m(ki)-172 b(\030)130 b(A.,)24 b Fb(A)i(me)-5 b(asur)g(e)25 b(the)-5 b(ory)25 b(the)-5 b(or)g(em)25 b(which)f(violates)g(the)h(T)-7 b(ertium)528 2370 y(non)35 b(Datur)p Fq(\(preprin)m(t)d(,)h(August)g(\(2001\)\).)328 2574 y([27])48 b([M)s(\030)-46 b(a1])37 b(M)s(\030)-46 b(adrec)m(ki)37 b(A.,)h Fb(On)g(Sazonov)f(typ)-5 b(e)39 b(top)-5 b(olo)g(gy)38 b(in)g(p-adic)g(Banach)f(sp)-5 b(ac)g(e)7 b Fq(,)528 2694 y(Math.)33 b(Zeitsc)m(hrift)f(30\(1985\),)f (225-236.)328 2897 y([28])48 b([M)s(\030)-46 b(a])42 b(M)s(\030)-46 b(adrec)m(ki)42 b(Andrzej.,)i Fb(A)f(pr)-5 b(o)g(of)42 b(of)h(the)g(R)n(iemann)e(hyp)-5 b(othesis)8 b Fq(,)43 b(preprin)m(t)528 3018 y(\(1997\).)328 3221 y([29])48 b([M)s(\030)-46 b(a2])41 b(M)s(\030)-46 b(adrec)m(ki)41 b(A.,)h Fb(A)g(pr)-5 b(o)g(of)41 b(of)h(the)g(R)n(iemann)e(hyp)-5 b(othesis)41 b(on)g(the)h(gr)-5 b(ound)528 3342 y(of)47 b(Br)-5 b(ouwer)47 b(lo)-5 b(gic)46 b(\(The)h(Ho)-5 b(dge)47 b(me)-5 b(asur)g(e)46 b(and)h(de)-5 b(c)g(omp)g(ositions)45 b(appr)-5 b(o)g(ach\))t Fq(,)528 3462 y(preprin)m(t)33 b(\(1998\).)328 3665 y([30])48 b([M)s(\030)-46 b(a3])42 b(M)s(\030)-46 b(adrec)m(ki)43 b(A.,)i Fb(A)f(short)f(c)-5 b(onstructive)44 b(maths)f(pr)-5 b(o)g(of)43 b(of)g(the)g(R)n(iemann) 528 3786 y(hyp)-5 b(othesis)48 b(\(The)g(appr)-5 b(o)g(ach)47 b(b)-5 b(ase)g(d)48 b(on)g(abstr)-5 b(act)48 b(Wiener)g(sp)-5 b(ac)g(es\))t Fq(,)50 b(preprin)m(t)528 3906 y(\(2000\).)328 4110 y([31])e([Mo])33 b(Mora)m(yne)h(M.,)f Fb(On)i(the)f(chr)-5 b(omatic)34 b(numb)-5 b(er)35 b(of)f(plane)7 b Fq(,)33 b(preprin)m(t)f(\(2000\).)328 4313 y([32])48 b([Na])36 b(Narkiewicz)g(W.,)h Fb(Elementary)h(and)f(analytic)h(the)-5 b(ory)38 b(of)f(algebr)-5 b(aic)37 b(num-)528 4433 y(b)-5 b(ers)41 b Fq(Monogra\014e)33 b(Matemat)m(yczne)h(\(MM)f(57\),)f(PWN,)h (W)-8 b(arsza)m(w)m(a)34 b(,1974.)328 4637 y([33])48 b([P)m(as52])e(Probabilit)m(y)d(Abstract)i(Servise)h(52)e(of)h(Bass,)j (Burdzy)-8 b(,)50 b(Sharp)s(e)45 b(and)528 4757 y(Susank)-5 b(a,)50 b Fb(A)d(pr)-5 b(o)g(of)46 b(of)h(the)g(R)n(iemann)e(hyp)-5 b(othesis)53 b Fq(\(Abstract)46 b(No)f(1324\),)j(31)528 4878 y(Aug)33 b(1999.)1898 5214 y(44)p eop %%Page: 45 45 45 44 bop 328 631 a Fq([34])48 b([P)m(e])37 b(P)m(enrose)g(R.,)e Fb(The)i(lar)-5 b(ge,)37 b(the)g(smal)5 b(l)37 b(and)f(the)i(human)e (mind)10 b Fq(,)36 b(Press)g(Syn-)528 751 y(dicate)d(of)f(theUniv)m (ersit)m(y)i(of)e(Cam)m(bridge,)g(1997.)328 955 y([35])48 b([Pi])201 b(Pitk)-5 b(anen)202 b(M.,)243 b Fb(Pr)-5 b(o)g(of)190 b(of)g(R)n(iemann)e(hyp)-5 b(othesis)8 b Fq(,)528 1075 y(preprin)m(t:arXiv:math.GM/0102031)30 b(5F)-8 b(eb\(2001\).)328 1279 y([36])48 b([P)m(o])34 b(P)m(oincare)e(H.,)h Fb(Scienc)-5 b(e)34 b(et)h(metho)-5 b(de)7 b Fq(,)32 b(P)m(aris)h(1927.)328 1482 y([37])48 b([Ric])32 b(Rice)h(H.G.,)f Fb(R)-5 b(e)g(cursive)34 b(r)-5 b(e)g(al)35 b(numb)-5 b(ers)8 b Fq(,P)-8 b(AMS)33 b(5\(1954\),)f(784-791.)328 1685 y([38])48 b([Rie])22 b(Riemann)f(B.,)p Fb(Ueb)-5 b(er)26 b(die)f(A)n(nzahl)g(der)g (Primzahlen)f(unter)i(einer)f(ge)-5 b(geb)g(ener)528 1806 y(Gr)g(osse)7 b Fq(,)66 b(Monatsb)s(eric)m(h)m(te)59 b(Ak)-5 b(ad.)59 b(Berlin,)64 b(No)m(v)m(em)m(b)s(er)59 b(1859=Gesammelte)528 1926 y(math.)32 b(W)-8 b(erk)m(e,)34 b(Berlin)e(1892,)f(145-153.)328 2130 y([39])48 b([Se1])34 b(Serre)g(J.P)-8 b(.,)35 b Fb(Quelques)g(applic)-5 b(ations)34 b(de)h(the)-5 b(or)g(eme)35 b(de)g(densite)g(de)h(Cheb-)528 2250 y(otar)-5 b(ev)11 b Fq(,)33 b(Publ.Math.)f(IHES)i(54\(1981\),)d (123-201.)328 2453 y([40])48 b([Se2])32 b(Serre)f(J.P)-8 b(.,)32 b Fb(A)n(nalo)-5 b(gues)32 b(kahleriens)g(de)h(c)-5 b(ertaines)32 b(c)-5 b(onje)g(ctur)g(es)33 b(de)g(Weil)g Fq(,)528 2574 y(Ann.)h(of)e(Mat.)g(71\(1960\),392-394.)328 2777 y([41])48 b([Sp])24 b(Sp)s(ec)m(k)m(er)i(E.,)g Fb(Nicht)g (konstructiv)h(b)-5 b(eweisb)g(ar)g(e)25 b(Satze)h(der)g(A)n(nalysis)8 b Fq(,)25 b(J.)e(Sym-)528 2897 y(b)s(olic)31 b(Logic)h(14\(1949\),)f (145-158.)328 3126 y(e-mail)f(:)43 b(madrec)m(ki@im.p)m(wr.wro)s(c.pl) 1898 5214 y(45)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF