Linear independence of odd zeta values using Siegel's lemma

Abstract We prove that among 1 and the odd zeta values , , , , at least are linearly independent over the rationals, for any sufficiently large odd integer . This is the first asymptotic improvement on the lower bound, logarithmic in , obtained by Ball–Rivoal in 2001. The proof is based on Siegel's lemma to construct non‐explicit linear forms in values at odd integers of the Riemann zeta function, instead of using explicit well‐poised hypergeometric series. A new refinement of Siegel's linear independence criterion is applied, together with a multiplicity estimate (namely a generalization of Shidlovsky's lemma). The result is also adapted to deal with values of the first polylogarithms at a fixed algebraic point in the unit disk, improving bounds of Rivoal and Marcovecchio.