{"paper":{"title":"When the number of divisors is a quadratic residue","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Olivier Bordell\\`es","submitted_at":"2017-01-09T18:16:13Z","abstract_excerpt":"Let $q > 2$ be a prime number and define $\\lambda_q := \\left( \\frac{\\tau}{q} \\right)$ where $\\tau(n)$ is the number of divisors of $n$ and $\\left( \\frac{\\cdot}{q} \\right)$ is the Legendre symbol. When $\\tau(n)$ is a quadratic residue modulo $q$, then $\\left( \\lambda_q \\star \\mathbf{1} \\right) (n)$ could be close to the number of divisors of $n$. This is the aim of this work to compare the mean value of the function $\\lambda_q \\star \\mathbf{1}$ to the well known average order of $\\tau$. The proof reveals that the results depend heavily on the value of $\\left( \\frac{2}{q} \\right)$. A bound for s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02286","kind":"arxiv","version":1},"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"}