Algorithms for the Hurwitz zeta function

Euler-Maclaurin summation

The Euler-Maclaurin formula allows evaluating the Hurwitz zeta function and its derivatives for general complex input. The algorithm is described in [Joh2013].

Parameter Taylor series

To evaluate ζ(s,a) for several nearby parameter values, the following Taylor expansion is useful:

ζ(s,a+x)=k=0(x)k(s)kk!ζ(s+k,a)

We assume that a1 is real and that σ=re(s) with K+σ>1. The tail is bounded by

k=K|x|k|(s)k|k!ζ(σ+k,a)k=K|x|k|(s)k|k![1aσ+k+1(σ+k1)aσ+k1].

Denote the term on the right by T(k). Then

|T(k+1)T(k)|=|x|a(k+σ1)(k+σ)(k+σ+a)(k+σ+a1)|k+s|(k+1)|x|a(1+1K+σ+a1)(1+|s1|K+1)=C

and if C<1,

k=KT(k)T(K)1C.