At least two of $\zeta(5),\zeta(7),\ldots,\zeta(35)$ are irrational

TL;DR

This paper proves that at least two of the zeta values at odd integers between 5 and 35 are irrational, improving previous bounds, and also establishes the irrationality of at least one beta value among certain even integers.

Contribution

It advances the understanding of the irrationality of zeta and beta values by lowering the known bounds for irrationality from 69 to 35 for zeta values.

Findings

At least two of $\zeta(5),\zeta(7),\ldots,\zeta(35)$ are irrational.

At least one of $eta(2),eta(4),\ldots,eta(10)$ is irrational.

Abstract

Let $\zeta(s)$ be the Riemann zeta function. We prove the statement in the title, which improves a recent result of Rivoal and Zudilin by lowering $69$ to $35$. We also prove that at least one of $\beta(2),\beta(4),\ldots,\beta(10)$ is irrational, where $\beta(s) = L(s,\chi_4)$ and $\chi_4$ is the Dirichlet character with conductor $4$.