{"paper":{"title":"Primes in short arithmetic progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dimitris Koukoulopoulos","submitted_at":"2014-05-26T14:46:24Z","abstract_excerpt":"Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\\le Q$ and for most positive real numbers $y\\le x$, every reduced arithmetic progression $a\\mod q$ has approximately the expected number of primes $p$ from the interval $(y,y+h]$, provided that $h>x^{1/6+\\epsilon}$ and $Q$ satisfies appropriate bounds in terms of $h$ and $x$. Moreover, we prove that, for most moduli $q\\le Q$ and for most positive real numbers $y\\le x$, there is at least one prime $p\\in(y,y+h]$ lying in every reduced arithmetic progression $a\\mod q$, provided that $1\\le Q^2\\le h/x^{1/15+\\epsilon}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.6592","kind":"arxiv","version":2},"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"}