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This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens' conjecture by showing that lim sup x→∞ M (x)x −1/2 ≥ 1.6383, and lim infx→∞ M (x)x −1/2 ≤ −1.6383.doi:10.1090/s0025-5718-2014-02916-5 fatcat:25k46oxgjfhlpkk3phexr2imoy