{"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/counting-zeros-of-dirichlet-l-functions","title":"Counting Zeros of Dirichlet $L$-Functions","arxiv_id":"2005.02989","date":"2020-05-06","proceeding":null,"authors":["Michael A. Bennett","Greg Martin","Kevin O'Bryant","Andrew Rechnitzer"],"abstract":"We give explicit upper and lower bounds for $N(T,\\chi)$, the number of zeros of a Dirichlet $L$-function with character $\\chi$ and height at most $T$. Suppose that $\\chi$ has conductor $q>1$, and that $T\\geq 5/7$. If $\\ell=\\log\\frac{q(T+2)}{2\\pi}> 1.567$, then \\begin{equation*} \\left| N(T,\\chi) - \\left( \\frac{T}{\\pi} \\log\\frac{qT}{2\\pi e} -\\frac{\\chi(-1)}{4}\\right) \\right| \\le 0.22737 \\ell + 2 \\log(1+\\ell) - 0.5. \\end{equation*} We give slightly stronger results for small $q$ and $T$. Along the way, we prove a new bound on $|L(s,\\chi)|$ for $\\sigma<-1/2$.","url_abs":"https://arxiv.org/abs/2005.02989v1","url_pdf":"https://arxiv.org/pdf/2005.02989v1.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":"counting-zeros-of-dirichlet-l-functions","repo_url":"https://github.com/truculentmath/DirichletCharacters","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}