{"paper":{"title":"On a pair of zeta functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2012-04-30T16:26:38Z","abstract_excerpt":"Let $m$ be a positive integer, and define $$\\zeta_m(s)=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}{n^s}\\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\zeta^*_m(s)=\\sum_{n=1}^\\infty\\frac{(-e^{2\\pi i/m})^{\\Omega(n)}}{n^s},$$ for $\\Re(s)>1$, where $\\omega(n)$ denotes the number of distinct prime factors of $n$, and $\\Omega(n)$ represents the total number of prime factors of $n$ (counted with multiplicity). In this paper we study these two zeta functions and related arithmetical functions. We show that $$\\sum^\\infty_{n=1\\atop n\\ \\text{is squarefree}}\\frac{(-e^{2\\pi i/m})^{\\omega(n)}}n=0\\quad\\text{if}\\ \\ m>4,$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.6689","kind":"arxiv","version":13},"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"}