Many values of the Riemann zeta function at odd integers are irrational

In this note, we announce the following result: at least 2(1-e)[formula could not be replicated] 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 improves on the lower bound [formula could not be replicated] log s that follows from the Ball-Rivoal theorem. We give the main ideas of the proof, which is based on an elimination process between several linear forms in odd zeta values with related coefficients.