{"id":"mertens-conjecture","domain":"named-results","type":"Disproven Conjecture","title":"Mertens Conjecture","slug":"mertens-conjecture","url":"/mathematics/named-results/mertens-conjecture/","posed":"1897, Franz Mertens","status":"disproven","disproved":"1985, Andrew Odlyzko and Herman te Riele","summary":"Conjectured |M(n)|<sqrt(n) for the Mertens function. Disproven 1985 via analytic methods, without ever finding an explicit counterexample. Would have implied the Riemann Hypothesis.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Odlyzko, A.M. and te Riele, H.J.J. (1985). Disproof of the Mertens conjecture. Journal für die reine und angewandte Mathematik, 357, 138-160."},{"tier":1,"citation":"Mertens, F. (1897). Über eine zahlentheoretische Funktion. Sitzungsberichte der Österreichischen Akademie der Wissenschaften, 106, 761-830."},{"tier":2,"citation":"Pintz, J. (1987). An effective disproof of the Mertens conjecture. Astérisque, 147-148, 325-333."},{"tier":3,"citation":"Derbyshire, J. (2003). Prime Obsession."}],"relationships":[{"type":"would_have_implied","target":"riemann-hypothesis","label":"Riemann Hypothesis"},{"type":"built_on","target":"prime-number","label":"Prime Number — Möbius function"},{"type":"tracked_in","target":"proof-status","label":"Proof Status Board"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.4"}