Many odd zeta values are irrational

Abstract

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

14 pages