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

Abstract

We consider computing the Riemann zeta function ζ(s) and Dirichlet L-functions L(s, χ) 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 nth Bernoulli number or the nth Euler number exactly.