{"paper":{"title":"The $r$th moment of the divisor function: an elementary approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Florian Luca, L\\'aszl\\'o T\\'oth","submitted_at":"2017-03-26T08:39:01Z","abstract_excerpt":"Let $\\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \\sum_{n\\le x} \\tau(n)^r =xC_{r} (\\log x)^{2^r-1}+O(x(\\log x)^{2^r-2}), $$ for any integer $r\\ge 2$. Here, $$ C_{r}=\\frac{1}{(2^r-1)!} \\prod_{p\\ge 2}\\left( \\left(1-\\frac{1}{p}\\right)^{2^r} \\left(\\sum_{\\alpha\\ge 0} \\frac{(\\alpha+1)^r}{p^{\\alpha}}\\right)\\right). $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.08785","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"}