Many odd zeta values are irrational

Overview & Synopsis

Building upon ideas of the second and third authors, we prove that at least 2^{(1−ε)(log s)/(log log s)} values of the Riemann zeta function at odd integers between 3 and s are irrational, where ε is any positive real number and s is large enough in terms of ε. This lower bound is asymptotically larger than any power of log s; it improves on the bound (1−ε) (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.