Linear independence of odd zeta values using Siegel's lemma

TL;DR

The paper advances the study of linear independence among odd zeta-values by introducing a non-explicit, Siegel-lemma–driven construction of rational linear forms, combined with a refined Siegel linear independence criterion and a generalized Shidlovsky multiplicity estimate. By carefully orchestrating derivatives and polylogarithm relations around z=−1, the authors obtain asymptotically sharp bounds that improve Ball–Rivoal’s logarithmic lower bound to a subpolynomial, specifically 0.21√s/√log s, for large odd s. The method also extends to polylogarithms at an algebraic point, yielding analogous independence results with explicit constants. This approach broadens the toolkit for irrationality and independence results in zeta-values and related special function values via non-explicit Diophantine constructions and differential-system analysis.

Abstract

We prove that among 1 and the odd zeta values $ζ(3)$, $ζ(5)$, \ldots, $ζ(s)$, at least $ 0.21 \sqrt{s}/\sqrt{\log s}$ are linearly independent over the rationals, for any sufficiently large odd integer $s$. This is the first asymptotic improvement on the lower bound, logarithmic in $s$, 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 $s$ polylogarithms at a fixed algebraic point in the unit disk, improving bounds of Rivoal and Marcovecchio.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Lemma 1
• Proposition 1
• proof : Proof of Proposition \ref{['propsiegel']}
• Theorem 3
• Theorem 4
• Proposition 2
• Lemma 2
• Lemma 3