{"paper":{"title":"Series with summands involving harmonic numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2022-10-13T17:59:14Z","abstract_excerpt":"For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\\sum_{0<k\\le n}\\frac1{k^m}\\ \\ (n=0,1,2,\\ldots).$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For example, we conjecture that $$\\sum_{k=0}^\\infty(6k+1)\\frac{\\binom{2k}k^3}{256^k}\\left(H_{2k}^{(3)}-\\frac{7}{64}H_{k}^{(3)}\\right)\n  =\\frac{25\\zeta(3)}{8\\pi}-G,$$ where $G$ denotes the Catalan constant $\\sum_{k=0}^\\infty(-1)^k/(2k+1)^2$. This paper contains $70$ conjectures posed by the author during 2022--2023."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.07238","kind":"arxiv","version":9},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2210.07238/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}