Rapid computation of special values of Dirichlet $L$-functions
| hal.structure.identifier | Lithe and fast algorithmic number theory [LFANT] | |
| hal.structure.identifier | Analyse cryptographique et arithmétique [CANARI] | |
| dc.contributor.author | JOHANSSON, Fredrik | |
| dc.date.accessioned | 2024-04-04T02:44:55Z | |
| dc.date.available | 2024-04-04T02:44:55Z | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/191431 | |
| dc.description.abstractEn | We 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.sponsorship | Sûreté numérique pour les preuves assistées par ordinateur - ANR-20-CE48-0014 | |
| dc.language.iso | en | |
| dc.title.en | Rapid computation of special values of Dirichlet $L$-functions | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Théorie des nombres [math.NT] | |
| dc.subject.hal | Informatique [cs]/Analyse numérique [cs.NA] | |
| dc.subject.hal | Mathématiques [math]/Analyse classique [math.CA] | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-03386620 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-03386620v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=JOHANSSON,%20Fredrik&rft.genre=preprint |
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