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

Fredrik Johansson (1, 2)

1 LFANT - Lithe and fast algorithmic number theory
2 CANARI - Analyse cryptographique et arithmétique

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.

Domaines: Théorie des nombres [math.NT]; Analyse numérique [cs.NA]; Analyse classique [math.CA]

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