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

Fredrik Johansson 1 
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
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.
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https://hal.inria.fr/hal-03386620
Contributor : Fredrik Johansson Connect in order to contact the contributor
Submitted on : Wednesday, October 20, 2021 - 12:55:05 AM
Last modification on : Wednesday, February 2, 2022 - 3:54:37 PM
Long-term archiving on: : Friday, January 21, 2022 - 7:08:09 PM

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

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