{"paper":{"title":"Congruences Involving Multiple Harmonic Sums and Finite Multiple Zeta Values","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jianqiang Zhao","submitted_at":"2014-04-14T12:02:16Z","abstract_excerpt":"Let $p$ be a prime and ${\\mathfrak P}_p$ the set of positive integers which are prime to $p$. Recently, Wang and Cai proved that for every positive integer $r$ and prime $p>2$ $$ \\sum_{\\substack{i+j+k=p^r\\\\ i,j,k\\in{\\mathfrak P}_p}} \\frac1{ijk} \\equiv\n  -2p^{r-1} B_{p-3} \\pmod{p^r}, $$ where $B_{p-3}$ is the $(p-3)$-rd Bernoulli number. In this paper we prove the following analogous result: Let $n=2$ or $4$. Then for every positive integer $r\\ge n/2$ and prime $p>4$ $$ \\sum_{\\substack{i_1+\\cdots+i_n=p^r\\\\ i_1,\\dots,i_n\\in{\\mathfrak P}_p}} \\frac1{i_1i_2\\cdots i_n} \\equiv\n  -\\frac{n!}{n+1} p^{r}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.3549","kind":"arxiv","version":3},"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"}