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
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History & 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.
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^(\(10^{40}\)). The sharpest lesson in the directory: a conjecture can imply RH and still be FALSE.