{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"253cf39fb0110185d7d6b52b1532f4767c99c3eac64896b2fbf2262a08eada6f","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"058fdac30056a2e870fab38e88224bb9407d5341f773c2fd053704e1d60959c9","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.quantum-chaos.montgomery-pair","kind":"well-posed","literature_status":"open","n":"1","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 $1 - (\\operatorname{sin}(\\pi u)/(\\pi u))^2$ for all test functions; it is proven only for Fourier support in $\\mid\\alpha\\mid < 1$. Answer: a proof.","problem_ref":null,"references":"","settled_by":"A proof of $F(\\alpha) = 1$ for |alpha| >= 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":"ea13ab5f74a1c82c80da965948277a56b113b5df57a11f41c996b317accddd90"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"7c38309f426414da167c37e65e2a31d8af74151da3598d0e630974dfa0a19399","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"2e6561461c9526d71a1c3797f3138e4808b8f30b08d88e30dfffe7f4c4d9029e","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"ZOlEGQ_Ry-krzsjVayiwkP0w7dpHpZelnzp3sZw_oDV_xfjZhiAT5-Y32V-lxazQqj5zCyXsznjFB6K6N2keCw"},"schema":"pubphys.envelope/1"},"record_hash":"7c38309f426414da167c37e65e2a31d8af74151da3598d0e630974dfa0a19399","leaf_index":1649}