{"paper":{"title":"A congruence involving harmonic sums modulo $p^{\\alpha}q^{\\beta}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lirui Jia, Tianxin Cai, Zhongyan Shen","submitted_at":"2015-03-10T07:42:21Z","abstract_excerpt":"In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \\begin{equation*} Z(p^{r})\\equiv-2p^{r-1}B_{p-3} ~(\\bmod ~ p^{r}), \\end{equation*} where $ Z(n)=\\sum\\limits_{i+j+k=n\\atop{i,j,k\\in\\mathcal{P}_{n}}}\\frac{1}{ijk}$ and $\\mathcal{P}_{n}$ denote the set of positive integers which are prime to $n$. In this note, we obtain a congruence for distinct odd primes $p,~q$ and positive integers $\\alpha,~\\beta$, \\begin{equation*} Z(p^{\\alpha}q^{\\beta})\\equiv 2(2-q)(1-\\frac{1}{q^{3}})p^{\\alpha-1}q^{\\beta-1}B_{p-3}\\pmod{p^{\\alpha}} \\end{equation"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02798","kind":"arxiv","version":2},"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"}