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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 $1\\le Q^2\\le h/x^{1/15+\\epsilon}$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1405.6592","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-05-26T14:46:24Z","cross_cats_sorted":[],"title_canon_sha256":"9f5411beabd505a54c7e7ec6cce31543162c95129a10169f8beacc5e99e61ea5","abstract_canon_sha256":"92bb5a01bf0f8a2c349bd08529d75273211aa80a74529ce5a206699f46340a43"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:44.688307Z","signature_b64":"nYsArfe4khwgZyrL7ApTTEUKkIjz1CUUQ4Wu+O5BR+sUp7boY6HsXlhuurpppu+YglletP6tOEiawZiYdydDCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"deaeda2c2099db571d0037caaba8d63c2ac69b401611938fb9fdfd49c5b238de","last_reissued_at":"2026-05-18T00:42:44.687741Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:44.687741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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