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 $\zeta (s,a)$ for several nearby parameter values, the following Taylor expansion is useful:

$$\zeta (s,a+x)=\sum _{k=0}^{\mathrm{\infty }}(-x{)}^k\frac{(s{)}_k}{k!}\zeta (s+k,a)$$

We assume that $a\ge 1$ is real and that $\sigma =\mathrm{re}(s)$ with $K+\sigma >1$. The tail is bounded by

$$\sum _{k=K}^{\mathrm{\infty }}|x{|}^k\frac{|(s{)}_k|}{k!}\zeta (\sigma +k,a)\le \sum _{k=K}^{\mathrm{\infty }}|x{|}^k\frac{|(s{)}_k|}{k!}[\frac{1}{a^{\sigma +k}}+\frac{1}{(\sigma +k-1)a^{\sigma +k-1}}].$$

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

$$\left|\frac{T(k+1)}{T(k)}\right|=\frac{|x|}{a}\frac{(k+\sigma -1)}{(k+\sigma )}\frac{(k+\sigma +a)}{(k+\sigma +a-1)}\frac{|k+s|}{(k+1)}\le \frac{|x|}{a}(1+\frac{1}{K+\sigma +a-1})(1+\frac{|s-1|}{K+1})=C$$

and if $C<1$,

$$\sum _{k=K}^{\mathrm{\infty }}T(k)\le \frac{T(K)}{1-C}.$$