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Rapid computation of special values of Dirichlet $L$-functions

hal.structure.identifierLithe and fast algorithmic number theory [LFANT]
hal.structure.identifierAnalyse cryptographique et arithmétique [CANARI]
dc.contributor.authorJOHANSSON, Fredrik
dc.date.accessioned2024-04-04T02:44:55Z
dc.date.available2024-04-04T02:44:55Z
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/191431
dc.description.abstractEnWe consider computing the Riemann zeta function $\zeta(s)$ and Dirichlet $L$-functions $L(s,\chi)$ to $p$-bit accuracy for large $p$. Using the approximate functional equation together with asymptotically fast computation of the incomplete gamma function, we observe that $p^{3/2+o(1)}$ bit complexity can be achieved if $s$ is an algebraic number of fixed degree and with algebraic height bounded by $O(p)$. This is an improvement over the $p^{2+o(1)}$ complexity of previously published algorithms and yields, among other things, $p^{3/2+o(1)}$ complexity algorithms for Stieltjes constants and $n^{3/2+o(1)}$ complexity algorithms for computing the $n$th Bernoulli number or the $n$th Euler number exactly.
dc.description.sponsorshipSûreté numérique pour les preuves assistées par ordinateur - ANR-20-CE48-0014
dc.language.isoen
dc.title.enRapid computation of special values of Dirichlet $L$-functions
dc.typeDocument de travail - Pré-publication
dc.subject.halMathématiques [math]/Théorie des nombres [math.NT]
dc.subject.halInformatique [cs]/Analyse numérique [cs.NA]
dc.subject.halMathématiques [math]/Analyse classique [math.CA]
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-03386620
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-03386620v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=JOHANSSON,%20Fredrik&rft.genre=preprint


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