{"paper":{"title":"An asymptotic upper bound on prime gaps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.NT","authors_text":"Andr\\'e LeClair","submitted_at":"2015-06-10T15:25:12Z","abstract_excerpt":"The Cram\\'er-Granville conjecture is an upper bound on prime gaps, $g_n = p_{n+1} - p_n < \\cCramer \\, \\log^2 p_n$ for some constant $\\cCramer \\geq 1$. Using a formula of Selberg, we first prove the weaker summed version: $\\sum_{n=1}^N g_n < \\sum_{n=1}^N \\log^2 p_n$. In the remainder of the paper we investigate which properties of the fluctuations $\\fluc (x) = \\pi (x) - \\Li(x)$ would imply the Cram\\'er-Granville conjecture is true and present two such properties, one of which assumes the Riemann Hypothesis; however we are unable to prove these properties are indeed satisfied. We argue that the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03359","kind":"arxiv","version":3},"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"}