Numerical verification of Littlewood's bounds for $|L(1,\chi )|$

Abstract

Let $L(s,\chi )$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $\chi$ defined $\mathrm{mod}q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $|L(1,\chi )|$ using the values of Euler's $\Gamma$ function. We also introduce an alternative way of computing $\log \Gamma (x)$ and $\psi (x)={\Gamma }^{\prime }/\Gamma (x)$,$x\in (0,1)$. Using such algorithms we numerically verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan estimates on $|L(1,\chi )|$, where $\chi$ runs over the non trivial primitive Dirichlet characters $\mathrm{mod}q$, for every odd prime $q$ up to $10^7$. The programs used and the results here described are collected at the following address \url{http://www.math.unipd.it/~languasc/Littlewood_ineq.html}.

arXiv comment21 pages, 2 tables, 12 figures. One reference updated

References in corpus (3)

• Euler's constant: Euler's work and modern developments
• A fast algorithm to compute the Ramanujan-Deninger gamma-function and some number-theoretic applications
• Computation of the Kummer ratio of the class number for prime cyclotomic fields

Cited by in corpus (5)

• A fast algorithm to compute the Ramanujan-Deninger gamma-function and some number-theoretic applications
• Uniform effective estimates for $|L(1,\chi )|$
• A unified strategy to compute some special functions of number-theoretic interest
• The Kummer ratio of the relative class number for prime cyclotomic fields
• Numerical estimates on the Landau-Siegel zero and other related quantities