Many odd zeta values are irrational

Abstract

Building upon ideas of the second and third authors, we prove that at least$2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$values of the Riemann zeta function at odd integers between 3 and$s$are irrational, where$\unicode[STIX]{x1D700}$is any positive real number and$s$is large enough in terms of$\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of$\log s$; it improves on the bound$(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.