# 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.

## Abstract

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.

## Implementation

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

## Computational data

Small files:

- The imaginary part of the Riemann zeta zero $\rho_1$ to over 303000 digits, decimal representation [txt, 0.3 MB]
- The imaginary part of the Riemann zeta zero $\rho_1$ to over 303000 digits, internal representation $a2^b \pm c2^d$ [txt, 0.3 MB]
- The Stieltjes constants $\gamma_0 \ldots \gamma_{100000}$ rounded to 20 decimal digits [txt, 2.8 MB]
- The Keiper-Li coefficients $\lambda_0 \ldots \lambda_{100000}$ rounded to 20 decimal digits [txt, 2.1 MB]

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 $a2^b \pm c2^d$):

- The Stieltjes constants $\gamma_0 \ldots \gamma_{100000}$ with accuracy between roughly 37000 and 10000 digits, internal representation [txt.tar.bz2, 902 MB compressed, 2.0 GB decompressed]
- The Keiper-Li coefficients $\lambda_0 \ldots \lambda_{100000}$ with accuracy between roughly 33000 and 2900 digits, internal representation [txt.tar.bz2, 741 MB compressed, 1.7 GB decompressed]

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

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