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.
Categories: math.NA cs.NA math.CA math.NT