The problem
Let M(n) = Σ_{k≤n} μ(k) with μ the Möbius function. Claim: |M(n)| < √n for all n > 1. Its truth would imply the Riemann Hypothesis AND the simplicity of ζ's zeros — which is precisely why its fall mattered.
The Mertens Conjecture
historicHistory & significance
Stieltjes claimed a proof in 1885 (never materialised); Mertens' own tables supported the inequality into the thousands. It became THE test case for RH-adjacent claims.
Connected problems
- inspired by The Riemann Hypothesis
More in analytic number theory
- same area Oppermann's Conjecture
- same area Andrica's Conjecture
- same area Legendre's Conjecture
References
- primaryRead the primary source →
- sourceWikipedia — Riemann hypothesis
The resolution (computer-assisted proof)
Disproved June 1985 by Andrew Odlyzko and Herman te Riele — without finding a counterexample. High-precision evaluation of the inverse Mellin transform plus the Chudnovsky brothers' large-scale arithmetic showed the limsup exceeds 1.06√n and liminf drops below −1.009√n, killing the conjecture by existence-at-infinity. Subsequent work (Kotnik–van de Lune) verified no counterexample below \(10^16\); the actual first crossing may exceed e^(\(10^4\)0). The sharpest lesson in the directory: a conjecture can imply RH and still be FALSE.
How to check it: Pintz had made the first crossing effective with an explicit (astronomical) bound, so the hunt was on — but Odlyzko and te Riele won by pure existence, evaluating enough zeta zeros to force a violation somewhere unbounded. No crossing has ever been exhibited; the hunt continues past \(10^{16}\). Note the moral survives intact: Mertens implies RH, yet Mertens is false.
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