{"paper":{"title":"On the symmetry of primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Giovanni Coppola","submitted_at":"2010-09-29T11:06:17Z","abstract_excerpt":"We prove a kind of \"almost all symmetry\" result for the primes, i.e. we give non-trivial bounds for the \"symmetry integral\", say $I_{\\Lambda}(N,h)$, of the von Mangoldt function $\\Lambda(n)$ ($:= \\log p$ for prime-powers $n=p^r$, 0 otherwise). We get $I_{\\Lambda}(N,h)\\ll NhL^5+Nh^{21/20}L^2$, with $L:=\\log N$; as a Corollary, we bound non-trivially the Selberg integral of the primes, i.e. the mean-square of $\\sum_{x<n\\le x+h}\\Lambda(n)-h$, over $x\\in [N,2N]$, to get the \"Prime Number Theorem in almost all short intervals\" of (log-powers!) length $h\\ge L^{11/2+\\epsilon}$. We trust here in the i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.6121","kind":"arxiv","version":9},"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"}