{"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/new-characterizations-of-the-summatory","title":"Exact formulas for partial sums of the Möbius function expressed by partial sums weighted by the Liouville lambda function","arxiv_id":"2102.05842","date":"2021-02-11","proceeding":null,"authors":["Maxie Dion Schmidt"],"abstract":"The Mertens function, $M(x) := \\sum_{n \\leq x} \\mu(n)$, is defined as the summatory function of the classical M\\\"obius function. The Dirichlet inverse function $g(n) := (\\omega+1)^{-1}(n)$ is defined in terms of the shifted strongly additive function $\\omega(n)$ that counts the number of distinct prime factors of $n$ without multiplicity. The Dirichlet generating function (DGF) of $g(n)$ is $\\zeta(s)^{-1} (1+P(s))^{-1}$ for $\\Re(s) > 1$ where $P(s) = \\sum_p p^{-s}$ is the prime zeta function. We study the distribution of the unsigned functions $|g(n)|$ with DGF $\\zeta(2s)^{-1}(1-P(s))^{-1}$ and $C_{\\Omega}(n)$ with DGF $(1-P(s))^{-1}$ for $\\Re(s) > 1$. We establish formulas for the average order and variance of $\\log C_{\\Omega}(n)$ and prove a central limit theorem for the distribution of its values on the integers $n \\leq x$ as $x \\rightarrow \\infty$. Discrete convolutions of the partial sums of $g(n)$ with the prime counting function provide new exact formulas for $M(x)$.","url_abs":"https://arxiv.org/abs/2102.05842v8","url_pdf":"https://arxiv.org/pdf/2102.05842v8.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":"new-characterizations-of-the-summatory","repo_url":"https://github.com/maxieds/MertensFunctionComputations","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}