Many values of the Riemann zeta function at odd integers are irrational. (Beaucoup de valeurs aux entiers impairs de la fonction zêta de Riemann sont irrationnelles.) (English. Abridged French version) Zbl 1398.11109

Authors’ abstract: In this note, we announce the following result: at least \(2^{(1 - \varepsilon) \frac{\log s}{\log \log s}}\) values of the Riemann zeta function at odd integers between 3 and \(s\) are irrational, where \(\varepsilon\) is any positive real number and \(s\) is large enough in terms of \(\varepsilon\). This improves on the lower bound \(\frac{1 - \varepsilon}{1 + \log 2} \log s\) 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.


11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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