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:

\[\zeta(s,a+x) = \sum_{k=0}^{\infty} (-x)^k \frac{(s)_k}{k!} \zeta(s+k,a)\]

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

\[\sum_{k=K}^{\infty} |x|^k \frac{|(s)_k|}{k!} \zeta(\sigma+k,a) \le \sum_{k=K}^{\infty} |x|^k \frac{|(s)_k|}{k!} \left[ \frac{1}{a^{\sigma+k}} + \frac{1}{(\sigma+k-1) a^{\sigma+k-1}} \right].\]

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

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

and if \(C < 1\),

\[\sum_{k=K}^{\infty} T(k) \le \frac{T(K)}{1-C}.\]