Linear independence of odd zeta values using Siegel's lemma
Résumé
We prove that among 1 and the odd zeta values ζ(3), ζ(5),. .. , ζ(s), at least 0.21 \sqrt{s/ log s} are linearly independent over the rationals, for any sufficiently large odd integer s. This improves on the lower bound (1 − ε) (log s)/(1 + log 2) obtained by Ball-Rivoal in 2001. Up to the numerical constant 0.21, it gives as a corollary a new proof of the lower bound on the number of irrationals in this family proved in 2020 by Lai-Yu. The proof is based on Siegel's lemma to construct non-explicit linear forms in odd zeta values, instead of using explicit well-poised hypergeometric series. Siegel's linear independence criterion (instead of Nesterenko's) is applied, with a multiplicity estimate (namely a generalisation of Shidlovsky's lemma). The result is also adapted to deal with values of the first s polylogarithms at a fixed algebraic point in the unit disk, improving bounds of Rivoal and Marcovecchio.
Domaines
Théorie des nombres [math.NT]
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