Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

Author: Fredrik Johansson

Final submitted version (PDF)

In Numerical Algorithms, DOI: 10.1007/s11075-014-9893-1

arXiv preprint: http://arxiv.org/abs/1309.2877, published September 11, 2013.


We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $\zeta(s,a)$ for $s, a \in \mathbb{C}$, along with an arbitrary number of derivatives with respect to $s$, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.


The implementation of the Hurwitz zeta function is available as part of the open source Arb library.

Computational data

Small files:

Huge files (each line contains five integers n a b c d, separated by single spaces, such that $\gamma_n$ (respectively $\lambda_n$) is contained in the ball $a2^b \pm c2^d$):

Last updated July 31, 2014. Contact: fredrik.johansson@gmail.com.

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