{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"85470f5f0255e97585ae2133ff90348543ca20b9acd37a5666a5fdceb27894c3","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["7c38309f426414da167c37e65e2a31d8af74151da3598d0e630974dfa0a19399","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"503d5aea911b9c19d5ab91d810fa9d329e1b02240d2d0f89b9cc0aa83c941e35","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":["ai:gpt-5.5"],"external_id":"mathph.quantum-chaos.montgomery-pair","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"Montgomery showed that the spacing between Riemann zeros matches random-matrix theory for some range of distances, assuming the Riemann hypothesis. Extending this to all distances is open.","posed_since":"1973","precise":"Let $F(\\alpha)$ be Montgomery's pair correlation function of normalized zeros. Prove $F(\\alpha) = 1$ for $\\mid\\alpha\\mid \\ge 1$ (assuming RH), which gives the GUE pair correlation $1 - (\\sin(\\pi u)/(\\pi u))^2$ for all test functions; it is proven only for Fourier support in $\\mid\\alpha\\mid < 1$. Answer: a proof.","problem_ref":"7c38309f426414da167c37e65e2a31d8af74151da3598d0e630974dfa0a19399","references":"","settled_by":"A proof of $F(\\alpha) = 1$ for $\\mid \\alpha \\mid \\ge 1$, or of the pair correlation for test functions with Fourier support beyond $[-1, 1]$.","status_note":"","title":"Montgomery's GUE pair correlation of the Riemann zeros for all ranges","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"404104088e540ce816c96f053af84786320fd5925ae7e53191e0dec6ff7cc639","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"987238cb951810fb3e4570827d2eb8a611c4d1054d2f7229c4d3693d6a8ee1e9","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"NHzqZWYQZu3XPSpqRvBUJA5bqj-konl0iICf2XbqAwYM-jistDmOHAbP-NuWrufQUYsnpnOz-JXo7zSbNcBgAw"},"schema":"pubphys.envelope/1"},"record_hash":"404104088e540ce816c96f053af84786320fd5925ae7e53191e0dec6ff7cc639","leaf_index":3446}