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

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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