Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture

2016/10/26 by G. Cameron Hurst, Hurst, Greg, Gregory B. Hurst · 1 citation
Computer Science · Mathematics · #Coding theory and cryptography #Polynomial and algebraic computation #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.1610.08551

Abstract

The Mertens function is defined as M(x) = ∑_{n ≤ x} μ(n), where μ(n) is the Möbius function. The Mertens conjecture states |M(x)/√(x)| < 1 for x > 1, which was proven false in 1985 by showing \liminf M(x)/√(x) < -1.009 and \limsup M(x)/√(x) > 1.06. The same techniques used were revisited here with present day hardware and algorithms, giving improved lower and upper bounds of -1.837625 and 1.826054. In addition, M(x) was computed for all x ≤ 10¹⁶, recording all extrema, all zeros, and 10⁸ values sampled at a regular interval. Lastly, an algorithm to compute M(x) in O(x^2/3+ε) time was used on all powers of two up to 2⁷³.

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