{"paper":{"title":"Arithmetic theory of harmonic numbers (II)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Li-Lu Zhao, Zhi-Wei Sun","submitted_at":"2009-11-23T19:27:44Z","abstract_excerpt":"For $k=1,2,\\ldots$ let $H_k$ denote the harmonic number $\\sum_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime $p>3$ we have $$\\sum_{k=1}^{p-1}\\frac{H_k}{k2^k}\\equiv\\frac7{24}pB_{p-3}\\pmod{p^2},\\ \\ \\sum_{k=1}^{p-1}\\frac{H_{k,2}}{k2^k}\\equiv-\\frac 38B_{p-3}\\pmod{p},$$ and $$\\sum_{k=1}^{p-1}\\frac{H_{k,2n}^2}{k^{2n}}\\equiv\\frac{\\binom{6n+1}{2n-1}+n}{6n+1}pB_{p-1-6n}\\pmod{p^2}$$ for any positive integer $n<(p-1)/6$, where $B_0,B_1,B_2,\\ldots$ are Bernoulli numbers, and $H_{k,m}:=\\sum_{j=1}^k 1/j^m$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.4433","kind":"arxiv","version":8},"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"}