This article is cited in 5 papers

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

Wadim Zudilin

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

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.

Keywords: irrationality; zeta value; hypergeometric series.

MSC: 11J72; 11M06; 33C20

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

Language: English

DOI: 10.3842/SIGMA.2018.028

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