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Mathematics of Computation

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Computations of the Mertens function and improved bounds on the Mertens conjecture

Author: Greg Hurst
Journal: Math. Comp. 87 (2018), 1013-1028
MSC (2010): Primary 11A25, 11M26, 11Y35, 11Y55, 11Y70
Published electronically: November 1, 2017
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Abstract: The Mertens function is defined as $ M(x) = \sum _{n \leq x} \mu (n)$, where $ \mu (n)$ is the Möbius function. The Mertens conjecture states $ \vert M(x)/\sqrt {x}\vert < 1$ for $ x > 1$, which was proven false in 1985 by showing $ \liminf M(x)/\sqrt {x} < -1.009$ and $ \limsup M(x)/\sqrt {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 \leq 10^{16}$, recording all extrema, all zeros, and $ 10^8$ values sampled at a regular interval. Finally, an algorithm to compute $ M(x)$ in $ O(x^{2/3+\varepsilon })$ time was used on all powers of two up to $ 2^{73}$.

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Greg Hurst
Affiliation: Wolfram Research Inc., 100 Trade Center Drive, Champaign, Illinois 61820

Keywords: The Mertens function, the Mertens conjecture
Received by editor(s): October 28, 2016
Received by editor(s) in revised form: March 4, 2017
Published electronically: November 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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