{"paper":{"title":"New upper bounds for Ramanujan primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anitha Srinivasan, Pablo Ar\\'es","submitted_at":"2017-06-22T10:22:17Z","abstract_excerpt":"For $n\\ge 1$, the $n^{\\rm th}$ Ramanujan prime is defined as the smallest positive integer $R_n$ such that for all $x\\ge R_n$, the interval $(\\frac{x}{2}, x]$ has at least $n$ primes. We show that for every $\\epsilon>0$, there is a positive integer $N$ such that if $\\alpha=2n\\left(1+\\dfrac{\\log 2+\\epsilon}{\\log n+j(n)}\\right)$, then $R_n< p_{[\\alpha]}$ for all $n>N$, where $p_i$ is the $i^{\\rm th}$ prime and $j(n)>0$ is any function that satisfies $j(n)\\to \\infty$ and $nj'(n)\\to 0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.07241","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"}