{"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/a-proof-to-the-riemann-hypothesis-using-a","title":"A Proof of the Riemann Hypothesis Using Bombieri's Equivalence Theorem","arxiv_id":"2108.02851","date":"2021-08-01","proceeding":null,"authors":["Xiao Lin"],"abstract":"The Riemann Hypothesis asserts that the Riemann $\\xi(s)$ function has no zeros in the critical strip $0<{\\rm Re}(s)<1$ except on the critical line ${\\rm Re}(s)=\\frac12$. Bombieri, in the official description of the Millennium Prize Problems, stated that the Riemann Hypothesis is equivalent to the condition that all local maxima of $\\xi(t)$ on the critical line are positive and all local minima are negative. In this paper, we pursue this criterion. We first show that $\\xi(s)$, when restricted to the critical line, satisfies a special differential equation, which ensures that it satisfies Bombieri's condition. Since a published proof of the sufficiency direction of Bombieri's theorem appears to be unavailable, we supply an independent proof of this implication. Using the Cauchy--Riemann equations, we prove that Bombieri's condition forces $\\xi(s)$ to have no zeros off the critical line. The Riemann Hypothesis follows. We also discuss P\\'olya's counterexample and other known counterexamples among L-functions, and demonstrate that none of them invalidates our approach.","url_abs":"https://arxiv.org/abs/2108.02851v20","url_pdf":"https://arxiv.org/pdf/2108.02851v20.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":"a-proof-to-the-riemann-hypothesis-using-a","repo_url":"https://github.com/linxiao19570629/rh","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}