{"paper":{"title":"The Shanks-R\\'enyi prime number race with many contestants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Youness Lamzouri","submitted_at":"2011-08-26T15:59:15Z","abstract_excerpt":"Under certain plausible assumptions, M. Rubinstein and P. Sarnak solved the Shanks--R\\'enyi race problem, by showing that the set of real numbers $x\\geq 2$ such that $\\pi(x;q,a_1)>\\pi(x;q,a_2)>...>\\pi(x;q,a_r)$ has a positive logarithmic density $\\delta_{q;a_1,...,a_r}$. Furthermore, they established that if $r$ is fixed, $\\delta_{q;a_1,...,a_r}\\to 1/r!$ as $q\\to \\infty$. In this paper, we investigate the size of these densities when the number of contestants $r$ tends to infinity with $q$. In particular, we deduce a strong form of a recent conjecture of A. Feuerverger and G. Martin which stat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5342","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"}