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

TL;DR

This paper introduces a faster method for computing special values of Dirichlet L-functions and related constants with near-optimal bit complexity, improving previous algorithms significantly.

Contribution

It presents a novel algorithm achieving $p^{3/2+o(1)}$ bit complexity for computing $ ext{zeta}$ and $L$-functions, surpassing earlier $p^{2+o(1)}$ methods.

Findings

Achieves $p^{3/2+o(1)}$ bit complexity for large $p$

Provides efficient algorithms for Stieltjes, Bernoulli, and Euler constants

Utilizes approximate functional equations and fast incomplete gamma function computations

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.