Rapid computation of special values of Dirichlet $L$-functions - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Preprints, Working Papers, ... Year : 2021

Rapid computation of special values of Dirichlet $L$-functions

Abstract

We consider computing the Riemann zeta function $\zeta(s)$ and Dirichlet $L$-functions $L(s,\chi)$ to $p$-bit accuracy for large $p$. Using the approximate functional equation together with asymptotically fast computation of the incomplete gamma function, we observe that $p^{3/2+o(1)}$ bit complexity can be achieved if $s$ is an algebraic number of fixed degree and with algebraic height bounded by $O(p)$. This is an improvement over the $p^{2+o(1)}$ complexity of previously published algorithms and yields, among other things, $p^{3/2+o(1)}$ complexity algorithms for Stieltjes constants and $n^{3/2+o(1)}$ complexity algorithms for computing the $n$th Bernoulli number or the $n$th Euler number exactly.
Fichier principal
Vignette du fichier
dirichlet.pdf (266.52 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03386620 , version 1 (20-10-2021)

Identifiers

  • HAL Id : hal-03386620 , version 1

Cite

Fredrik Johansson. Rapid computation of special values of Dirichlet $L$-functions. 2021. ⟨hal-03386620⟩
316 View
167 Download

Share

Gmail Mastodon Facebook X LinkedIn More