{"paper":{"title":"Positive proportion of short intervals containing a prescribed number of primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daniele Mastrostefano","submitted_at":"2018-12-31T13:10:40Z","abstract_excerpt":"We will prove that for every $m\\geq 0$ there exists an $\\varepsilon=\\varepsilon(m)>0$ such that if $0<\\lambda<\\varepsilon$ and $x$ is sufficiently large in terms of $m$ and $\\lambda$, then $$|\\lbrace n\\leq x: |[n,n+\\lambda\\log n]\\cap \\mathbb{P}|=m\\rbrace|\\gg_{m,\\lambda} x.$$ The value of $\\varepsilon(m)$ and the implicit constant on $\\lambda$ and $m$ may be made explicit. This is an improvement of an author's previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when considering the parameters $m$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.11784","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"}