Many odd zeta values are irrational

Building upon ideas of the second and third authors, we prove that at least 2((1-epsilon)(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 (1 - epsilon)(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.