A note on the number of irrational odd zeta values

TL;DR

This paper establishes a new lower bound on the number of irrational odd zeta values up to a certain integer, improving previous bounds by employing an optimal zero distribution of auxiliary rational functions related to Euler's totient function.

Contribution

It provides a significantly improved lower bound on the count of irrational odd zeta values, using a novel zero distribution approach linked to Euler's totient function.

Findings

New lower bound for irrational odd zeta values

Improved over previous exponential bounds

Uses optimal zero distribution related to Euler's totient function

Abstract

It is proved that, for all odd integer $s \geqslant s_0(\varepsilon)$, there are at least $\big( c_0 - \varepsilon \big) \frac{s^{1/2}}{(\log s)^{1/2}} $ many irrational numbers among the following odd zeta values: $\zeta(3),\zeta(5),\zeta(7),\cdots,\zeta(s)$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. The work is based on the previous work of Fischler, Sprang and Zudilin [FSZ19], improves the lower bound $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ therein. The main new ingredient is an optimal design for the zeros of the auxiliary rational functions, which relates to the inverse of Euler totient funtion.