Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture
Hurst, Greg · arXiv (Cornell University) · 2016
The Mertens function is defined as $M(x) = \sum_{n \leq x} μ(n)$, where $μ(n)$ is the Möbius function. The Mertens conjecture states $|M(x)/\sqrt{x}| 1$, which was proven false in 1985 by showing $\liminf M(x)/\sqrt{x} 1.06$. 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. Lastly, an algorithm to compute $M(x)$ in $O(x^{2/3+\varepsilon})$ time was used on all powers of two up to $2^{73}$.