At least two of $\zeta (5),\zeta (7),\dots ,\zeta (35)$ 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 $\beta (2),\beta (4),\dots ,\beta (10)$ is irrational, where $\beta (s)=L(s,{\chi }_4)$ and ${\chi }_4$ is the Dirichlet character with conductor $4$.

14 pages. A better collection of parameters for Thm. 6.1 (motivated by one referee). Some minor corrections according to referee reports