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
DOI:
https://doi.org/10.1090/mcom/3401
Published electronically:
December 26, 2018
MathSciNet review:
3925487
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Abstract | References | Similar Articles | Additional Information
Abstract: The generalized Stieltjes constants are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function
about its unique pole
. 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
. 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
to 1000 digits in a minute for any
up to
. We also provide other interesting integral representations for
,
,
, 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
Email:
fredrik.johansson@gmail.com
Iaroslav V. Blagouchine
Affiliation:
SeaTech, University of Toulon, France; and Steklov Institute of Mathematics at St. Petersburg (Russian Academy of Sciences), Russia.
Email:
iaroslav.blagouchine@univ-tln.fr, iaroslav.blagouchine@pdmi.ras.ru
DOI:
https://doi.org/10.1090/mcom/3401
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