JOHANSSON, Fredrik; BLAGOUCHINE, Iaroslav

doi:10.1090/mcom/3401

Métadonnées

Afficher la notice complète

Licence d’utilisation du document

JOHANSSON, Fredrik
Lithe and fast algorithmic number theory [LFANT]

BLAGOUCHINE, Iaroslav
Université de Toulon - École d’ingénieurs SeaTech [UTLN SeaTech]

Langue

Article de revue

Ce document a été publié dans

Mathematics of Computation. 2019, vol. 88, n° 318

American Mathematical Society

Résumé en anglais

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.< Réduire

Mots clés en anglais

Complex integration

Integral representation

Riemann zeta function

Hurwitz zeta function

Complexity

Numerical integration

Stieltjes constants

Arbitrary-precision arithmetic

Rigorous error bounds

Métadonnées

Partager cette publication !

Licence d’utilisation du document

Computing Stieltjes constants using complex integration

Langue

Ce document a été publié dans

Résumé en anglais

Mots clés en anglais

URI

DOI

Origine

Unités de recherche