{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"a5a7bae9396ec5e010d818ca30b4ecce9ba2d46c74f41b760cfddf214a63a4e1","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"30fb3f271026f3a0629dd11e526f3831f5e710e68029cfdbeaff8f27d7dca9a1","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"identification","assisted_by":[],"external_id":"mathph.quantum-chaos.hilbert-polya","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"If the Riemann zeros were the energy levels of a quantum system, they would all be real numbers, which is the Riemann hypothesis. The task is to find such a system.","posed_since":"","precise":"Find a self-adjoint operator $H$ on a Hilbert space, defined independently of the zeros, with spectrum $\\{\\gamma_n\\}$ where $\\zeta(1/2 + i \\gamma_n) = 0$, and prove the spectral identity. The Berry-Keating proposal $H = xp$ with suitable boundary conditions reproduces the smooth counting function $N(E) \\sim (E/(2 \\pi)) \\operatorname{log}(E/(2 \\pi e)) + 7/8$ but not the individual zeros. Answer: an explicit operator with proof.","problem_ref":null,"references":"","settled_by":"A proof that the spectrum of an explicitly constructed self-adjoint operator equals the set of nontrivial zeros.","status_note":"Connes and collaborators proposed trace-formula and prolate-operator constructions (1998 to 2026) whose low spectra approach the zeros numerically; no proof of the identity.","title":"A self-adjoint operator whose eigenvalues are the Riemann zeros","topic_ref":"ea13ab5f74a1c82c80da965948277a56b113b5df57a11f41c996b317accddd90"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"1232c36fa3b984b9f7e8b0fcea61df77159a03e98ae63ab6c1ea1036aaf3955c","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"bedf983b5c2593e7e3e55cd39fd899a4117faebb31121b343a82f9c06fa2f970","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"uLvO-Jq_daI-IprQGxNKbSTC1Z6izCj-poOGm4djrv52b4Kvu8TkbwNmmkI8x0XlZDNmW2tHysfvwpxmQ99FAg"},"schema":"pubphys.envelope/1"},"record_hash":"1232c36fa3b984b9f7e8b0fcea61df77159a03e98ae63ab6c1ea1036aaf3955c","leaf_index":1648}