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

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 $|M(x)/\sqrt {x}| < 1$ 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 $2^{73}$.