{"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/computation-of-the-totient-summatory-function","title":"Computation of the Totient Summatory Function","arxiv_id":"2506.07386","date":"2025-06-09","proceeding":null,"authors":["Lucas Augustus Brown"],"abstract":"An algorithm is devised for computing $\\Phi(n) = \\phi(1) + \\phi(2) + \\cdots + \\phi(n)$ in time $\\widetilde{\\Theta}(n^{2/3})$ and space $\\widetilde{\\Theta}(n^{1/3})$. The starting point is an existing algorithm based on the Dirichlet hyperbola method and the Mertens function. The algorithm is then used to compute $\\Phi(10^{19}) = 30396355092701331435065976498046398788$.","url_abs":"https://arxiv.org/abs/2506.07386v1","url_pdf":"https://arxiv.org/pdf/2506.07386v1.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":"computation-of-the-totient-summatory-function","repo_url":"https://github.com/lucasaugustus/totientsum","is_official":1,"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}