Rapid computation of special values of Dirichlet L-functions

10/20/2021

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by   Fredrik Johansson, et al.

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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.