{"paper":{"title":"Generalized Ramanujan Primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jonathan Sondow, Nadine Amersi, Olivia Beckwith, Ryan Ronan, Steven J. Miller","submitted_at":"2011-08-02T02:55:50Z","abstract_excerpt":"In 1845, Bertrand conjectured that for all integers $x\\ge2$, there exists at least one prime in $(x/2, x]$. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any $n\\ge1$, there is a (smallest) prime $R_n$ such that $\\pi(x)- \\pi(x/2) \\ge n$ for all $x \\ge R_n$. In 2009 Sondow called $R_n$ the $n$th Ramanujan prime and proved the asymptotic behavior $R_n \\sim p_{2n}$ (where $p_m$ is the $m$th prime). In the present paper, we generalize the interval of interest by introducing a parameter $c \\in (0,1)$ and defining the $n$th $c$-Ramanujan prime as "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0475","kind":"arxiv","version":4},"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"}