The aim of the author in this paper is to propose an algorithm for the fast evaluation of the Hurwitz zeta function \(\zeta(s,a)\), for integer values of \(s\) and algebraic values of \(a\). The paper follows the lines of previous work by the author (cited in the references), where algorithms for the fast evaluation of elementary and higher transcendental functions – as well as for the classical constants \(e,\pi\) and the Euler constant \(\gamma\) – have been presented. The general method is based on the fast evaluation for functions of the type of the Siegel \(E\)-function (the so-called FEE method), which has a complexity close to the best possible one. It is to be noted that an application of the FEE method to the fast evaluation of the Riemann zeta function \(\zeta(s)\) for integer values of the argument \(s\) was given by the author in [Probl. Inf. Transm. 31, 353-362 (1995; Zbl 0895.11032)] but that for fractional values of \(s\) no method for the fast evaluation of \(\zeta(s)\) has yet been found. The paper finishes with a theorem on the evaluation complexity of Dirichlet series \(L(s,\chi)\) for natural values of the argument, \(s=k\), \(k\geq 2\), and any Dirichlet character \(\chi(l)\) modulo \(m\), \(m\geq 2\), with \(m\) integer, for which the author finds the estimate \(s_L(n)= O(M(n)\log^2n)\).
This is the same kind of result as the one obtained for the Hurwitz zeta function under the conditions described. Remarkably, if coincides also with the result corresponding to the evaluation complexity for the above mentioned constants and for the elementary and higher transcendental functions, at algebraic values of the argument.