{"paper":{"title":"Average $r$-rank Artin Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Cihan Pehlivan, Lorenzo Menici","submitted_at":"2015-04-07T11:25:43Z","abstract_excerpt":"Let $\\Gamma\\subset\\mathbb{Q}^*$ be a finitely generated subgroup and let $p$ be a prime such that the reduction group $\\Gamma_p$ is a well defined subgroup of the multiplicative group $\\mathbb{F}_p^*$. We prove an asymptotic formula for the average of the number of primes $p\\le x$ for which the index $[\\mathbb{F}_p^*:\\Gamma_p]=m$. The average is performed over all finitely generated subgroups $\\Gamma=\\langle a_1,\\dots,a_r \\rangle\\subset\\mathbb{Q}^*$, with $a_i\\in\\mathbb{Z}$ and $a_i\\le T_i$ with a range of uniformity: $T_i>\\exp(4(\\log x \\log\\log x)^{\\frac{1}{2}})$ for every $i=1,\\dots,r$. We a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01554","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"}