{"paper":{"title":"Convolution sums of some functions on divisors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Heekyoung Hahn","submitted_at":"2015-07-16T00:57:06Z","abstract_excerpt":"One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\\widetilde{\\sigma}_s(n)=\\sum_{d|n}(-1)^{d-1}d^s$ and $\\widehat{\\sigma}_s(n)=\\sum_{d|n}(-1)^{\\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\\mathcal{P}(q)$, $\\mathcal{E}(q)$, and $\\mathcal{Q}(q)$ related to $\\widetilde{\\sigma}(n)$, $\\widehat{\\sigma}(n)$, and $\\widetilde{\\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04426","kind":"arxiv","version":1},"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"}