{"paper":{"title":"New congruences 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":"2014-07-31T15:53:32Z","abstract_excerpt":"Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k,\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)},\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where $H_k=\\sum_{0<j\\le k}1/j$ and $H_k^{(2)}=\\sum_{0<j\\le k}1/j^2$. In particular, we show that \\begin{gather*}\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k\\equiv(-1)^{\\langle a\\rangle_p}\\,2\\left(B_{p-1}(a)-B_{p-1}\\right)\\pmod p, \\\\\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)}\\equiv -E_{p-3}(a)\\pmod p, \\\\(2a-1)\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.8465","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1407.8465/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"}