Numerical Algorithms

, Volume 69, Issue 2, pp 253–270 | Cite as

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

Original Paper

Abstract

We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function ζ(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.

Keywords

Hurwitz zeta function Riemann zeta function Arbitrary-precision arithmetic Rigorous numerical evaluation Fast polynomial arithmetic Power series 

PAC Codes

65D20 68W30 33F05 11-04 11M06 11M35 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.RISCJohannes Kepler UniversityLinzAustria

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