{"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/exact-formulas-for-the-generalized-sum-of","title":"Exact Formulas for the Generalized Sum-of-Divisors Functions","arxiv_id":"1705.03488","date":"2017-05-09","proceeding":null,"authors":["Maxie D. Schmidt"],"abstract":"We prove new exact formulas for the generalized sum-of-divisors functions, $\\sigma_{\\alpha}(x) := \\sum_{d|x} d^{\\alpha}$. The formulas for $\\sigma_{\\alpha}(x)$ when $\\alpha \\in \\mathbb{C}$ is fixed and $x \\geq 1$ involves a finite sum over all of the prime factors $n \\leq x$ and terms involving the $r$-order harmonic number sequences and the Ramanujan sums $c_d(x)$. The generalized harmonic number sequences correspond to the partial sums of the Riemann zeta function when $r > 1$ and are related to the generalized Bernoulli numbers when $r \\leq 0$ is integer-valued. A key part of our new expansions of the Lambert series generating functions for the generalized divisor functions is formed by taking logarithmic derivatives of the cyclotomic polynomials, $\\Phi_n(q)$, which completely factorize the Lambert series terms $(1-q^n)^{-1}$ into irreducible polynomials in $q$. We focus on the computational aspects of these exact expressions, including their interplay with experimental mathematics, and comparisons of the new formulas for $\\sigma_{\\alpha}(n)$ and the summatory functions $\\sum_{n \\leq x} \\sigma_{\\alpha}(n)$. Keywords: divisor function; sum-of-divisors function; Lambert series; perfect number. MSC (2010): 30B50; 11N64; 11B83","url_abs":"http://arxiv.org/abs/1705.03488v4","url_pdf":"http://arxiv.org/pdf/1705.03488v4.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":"exact-formulas-for-the-generalized-sum-of","repo_url":"https://github.com/maxieds/ManuscriptComputationalData","is_official":0,"mentioned_in_paper":0,"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}