{"id":"riemann-hypothesis","domain":"named-results","also_in_domain":"number-theory","type":"Conjecture","title":"Riemann Hypothesis","slug":"riemann-hypothesis","url":"/mathematics/named-results/riemann-hypothesis/","statement":"Every non-trivial zero of the Riemann zeta function has real part exactly 1/2.","proposed":"1859, Bernhard Riemann","status":"unsolved","summary":"Millennium Prize Problem ($1M). Controls the precise distribution of prime numbers. Verified numerically for first 10^13 zeros; not proven.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie."},{"tier":1,"citation":"Titchmarsh, E.C. (1986). The Theory of the Riemann Zeta-Function. 2nd ed. Oxford University Press."},{"tier":2,"citation":"Bombieri, E. (2000). The Riemann Hypothesis. Clay Mathematics Institute.","url":"https://www.claymath.org/millennium-problems"},{"tier":3,"citation":"Derbyshire, J. (2003). Prime Obsession. Joseph Henry Press."}],"relationships":[{"type":"controls","target":"prime-number","label":"Prime Number distribution"},{"type":"millennium_prize","target":null,"label":"Clay Mathematics Institute, $1M"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.5"}