Computing Stieltjes constants using complex integration
| hal.structure.identifier | Lithe and fast algorithmic number theory [LFANT] | |
| dc.contributor.author | JOHANSSON, Fredrik | |
| hal.structure.identifier | Université de Toulon - École d’ingénieurs SeaTech [UTLN SeaTech] | |
| dc.contributor.author | BLAGOUCHINE, Iaroslav | |
| dc.date.accessioned | 2024-04-04T03:05:21Z | |
| dc.date.available | 2024-04-04T03:05:21Z | |
| dc.date.created | 2018 | |
| dc.date.issued | 2019 | |
| dc.identifier.issn | 0025-5718 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/193237 | |
| dc.description.abstractEn | 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. | |
| dc.language.iso | en | |
| dc.publisher | American Mathematical Society | |
| dc.subject.en | Complex integration | |
| dc.subject.en | Integral representation | |
| dc.subject.en | Riemann zeta function | |
| dc.subject.en | Hurwitz zeta function | |
| dc.subject.en | Complexity | |
| dc.subject.en | Numerical integration | |
| dc.subject.en | Stieltjes constants | |
| dc.subject.en | Arbitrary-precision arithmetic | |
| dc.subject.en | Rigorous error bounds | |
| dc.title.en | Computing Stieltjes constants using complex integration | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1090/mcom/3401 | |
| dc.subject.hal | Mathématiques [math]/Analyse classique [math.CA] | |
| dc.subject.hal | Informatique [cs]/Analyse numérique [cs.NA] | |
| dc.identifier.arxiv | 1804.01679 | |
| bordeaux.journal | Mathematics of Computation | |
| bordeaux.volume | 88 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 318 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-01758620 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01758620v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Mathematics%20of%20Computation&rft.date=2019&rft.volume=88&rft.issue=318&rft.eissn=0025-5718&rft.issn=0025-5718&rft.au=JOHANSSON,%20Fredrik&BLAGOUCHINE,%20Iaroslav&rft.genre=article |
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