Many odd zeta values are irrational

TL;DR

This paper proves that a super-polynomial number of odd zeta values are irrational, significantly improving previous bounds and advancing understanding of the irrationality of these special values.

Contribution

It establishes a new lower bound on the number of irrational odd zeta values, surpassing previous results by leveraging linear forms in zeta values.

Findings

Number of irrational odd zeta values grows faster than any power of log s

Improves upon the Ball--Rivoal bound for irrationality of zeta values

Uses construction of linear forms in odd zeta values

Abstract

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any positive real number and $s$ is large enough in terms of $\varepsilon$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $\frac{1-\varepsilon}{1+\log2}\log s$ that follows from the Ball--Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.