{"paper":{"title":"Exact Formulas for the Generalized Sum-of-Divisors Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maxie D. Schmidt","submitted_at":"2017-05-09T18:45:21Z","abstract_excerpt":"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.\n  A key part of our new expa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.03488","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}