At least two of $\zeta(5),\zeta(7),\ldots,\zeta(35)$ are 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 $4$.