A note on the number of irrational odd zeta values, II

TL;DR

This paper establishes a lower bound on the number of irrational values among certain odd zeta functions, improving previous results by a constant factor using advanced elimination and factor techniques.

Contribution

It introduces an improved lower bound on irrational zeta values among odd integers, combining recent elimination methods with classical factor techniques.

Findings

At least 1.284 * sqrt(s / log s) of the zeta values are irrational for large even s.

The result enhances previous bounds by a constant factor.

The proof integrates Fischler-Sprang-Zudilin's elimination technique with Zudilin's $\

Abstract

We prove that there are at least $1.284 \cdot \sqrt{s/\log s}$ irrational numbers among $\zeta(3)$, $\zeta(5)$, $\zeta(7)$, $\ldots$, $\zeta(s-1)$ for any sufficiently large even integer $s$. This result improves upon the previous finding by a constant factor. The proof combines the elimination technique of Fischler-Sprang-Zudilin (2019) with the $\Phi_n$ factor method of Zudilin (2001).