{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/an-introduction-to-the-bernoulli-function","title":"An introduction to the Bernoulli function","arxiv_id":"2009.06743","date":"2020-09-11","proceeding":null,"authors":["Peter H. N. Luschny"],"abstract":"We explore a variant of the zeta function interpolating the Bernoulli numbers based on an integral representation suggested by J. Jensen. The Bernoulli function $\\operatorname{B}(s, v) = - s\\, \\zeta(1-s, v)$ can be introduced independently of the zeta function if it is based on a formula first given by Jensen in 1895. We examine the functional equation of $\\operatorname{B}(s, v)$ and its representation by the Riemann $\\zeta$ and $\\xi$ function, and recast classical results of Hadamard, Worpitzky, and Hasse in terms of $\\operatorname{B}(s, v).$ The extended Bernoulli function defines the Bernoulli numbers for odd indices basing them on rational numbers studied by Euler in 1735 that underlie the Euler and Andr\\'{e} numbers. The Euler function is introduced as the difference between values of the Hurwitz-Bernoulli function. The Andr\\'{e} function and the Seki function are the unsigned versions of the extended Euler resp. Bernoulli function.","url_abs":"https://arxiv.org/abs/2009.06743v2","url_pdf":"https://arxiv.org/pdf/2009.06743v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"an-introduction-to-the-bernoulli-function","repo_url":"https://github.com/PeterLuschny/BernoulliFunction","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}