Irrationality of values of L-functions of Dirichlet characters

Abstract

In a recent paper with Sprang and Zudilin, the following result was proved: if $a$ is large enough in terms of $\epsilon >0$, then at least $2^{(1-\epsilon )\frac{\log a}{\log \log a}}$ values of the Riemann zeta function at odd integers between $3$ and $a$ are irrational. This improves on the Ball-Rivoal theorem, that provides only $\frac{1-\epsilon }{1+\log 2}\log a$ such irrational values -- but with a stronger property: they are linearly independent over the rationals.In the present paper we generalize this recent result to both $L$-functions of Dirichlet characters and Hurwitz zeta function. The strategy is different and less elementary: the construction is related to a Padé approximation problem, and a generalization of Shidlovsky's lemma is used to apply Siegel's linear independence criterion. We also improve the analogue of the Ball-Rivoal theorem in this setting: we obtain $\frac{1-\epsilon }{1+\log 2}\log a$ linearly independent values $L(f,s)$ with $s\le a$ of a fixed parity, when $f$ is a Dirichlet character. The new point here is that the constant $1+\log 2$ does not depend on $f$.

References in corpus (4)

• Many odd zeta values are irrational
• One of the Odd Zeta Values from $\zeta (5)$ to $\zeta (25)$ Is Irrational. By Elementary Means
• Infinitely many odd zeta values are irrational. By elementary means
• On the linear independence of the special values of a Dirichlet series with periodic coefficients

Cited by in corpus (4)

• A note on the number of irrational odd zeta values
• A Linear independence result for $p$-adic $L$-values
• Arithmetic of Catalan's constant and its relatives
• On linear independence of Dirichlet $L$ values