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Mathematics of Computation

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Computing Stieltjes constants using complex integration

Authors: Fredrik Johansson and Iaroslav V. Blagouchine
Journal: Math. Comp. 88 (2019), 1829-1850
MSC (2010): Primary 11M35, 65D20; Secondary 65G20
Published electronically: December 26, 2018
MathSciNet review: 3925487
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Abstract: The generalized Stieltjes constants $\gamma _n(v)$ are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function $\zeta (s,v)$ about its unique pole $s = 1$. In this work, we devise an efficient algorithm to compute these constants to arbitrary precision with rigorous error bounds, for the first time achieving this with low complexity with respect to the order $n$. Our computations are based on an integral representation with a hyperbolic kernel that decays exponentially fast. The algorithm consists of locating an approximate steepest descent contour and then evaluating the integral numerically in ball arithmetic using the Petras algorithm with a Taylor expansion for bounds near the saddle point. An implementation is provided in the Arb library. We can, for example, compute $\gamma _n(1)$ to 1000 digits in a minute for any $n$ up to $n=10^{100}$. We also provide other interesting integral representations for $\gamma _n(v)$, $\zeta (s)$, $\zeta (s,v)$, some polygamma functions, and the Lerch transcendent.

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Additional Information

Fredrik Johansson
Affiliation: LFANT project-team, INRIA, Institut de Mathématiques de Bordeaux, Bordeaux, France
MR Author ID: 999321

Iaroslav V. Blagouchine
Affiliation: SeaTech, University of Toulon, France; and Steklov Institute of Mathematics at St. Petersburg (Russian Academy of Sciences), Russia.
MR Author ID: 906772

Keywords: Stieltjes constants, Hurwitz zeta function, Riemann zeta function, integral representation, complex integration, numerical integration, complexity, arbitrary-precision arithmetic, rigorous error bounds.
Received by editor(s): May 30, 2018
Received by editor(s) in revised form: August 11, 2018
Published electronically: December 26, 2018
Article copyright: © Copyright 2018 American Mathematical Society