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

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.

References in corpus (3)

• Hypergeometry inspired by irrationality questions
• A note on odd zeta values
• Infinitely many odd zeta values are irrational. By elementary means

Cited by in corpus (6)

• The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants
• A note on the number of irrational odd zeta values
• Hypergeometry inspired by irrationality questions
• Irrationality of values of L-functions of Dirichlet characters
• Arithmetic of Catalan's constant and its relatives
• Some hypergeometric integrals for linear forms in zeta values