I report on a recent considerable improvement of the asymptotic lower bound on the number of irrational odd zeta values as originally given in the Ball–Rivoal theorem (2000). The proof is based on a new construction of several approximating linear forms, with related coefficients, in odd zeta values.

This is joint work with Stéphane Fischler (Paris, France) and Johannes Sprang (Regensburg, Germany).