### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 028, 8 pages      arXiv:1801.09895      https://doi.org/10.3842/SIGMA.2018.028
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

### One of the Odd Zeta Values from $\zeta(5)$ to $\zeta(25)$ Is Irrational. By Elementary Means

Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, The Netherlands

Received January 31, 2018, in final form March 26, 2018; Published online March 29, 2018

Abstract
Available proofs of result of the type 'at least one of the odd zeta values $\zeta(5),\zeta(7),\dots,\zeta(s)$ is irrational' make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are however counted as highly non-elementary, therefore leaving the partial irrationality result inaccessible to general mathematics audience in all its glory. Here we modify the original construction of linear forms in odd zeta values to produce, for the first time, an elementary proof of such a result — a proof whose technical ingredients are limited to the prime number theorem and Stirling's approximation formula for the factorial.

Key words: irrationality; zeta value; hypergeometric series.

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