• DOI:10.1090/mcom/3275
• Corpus ID: 6760210

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

@article{Hurst2016ComputationsOT,
  title={Computations of the Mertens function and improved bounds on the Mertens conjecture},
  author={Greg Hurst},
  journal={Math. Comput.},
  year={2016},
  volume={87},
  pages={1013-1028}
}

• Greg Hurst
• Published 26 October 2016
• Mathematics
• Math. Comput.

The Mertens function is defined as $M(x) = \sum_{n \leq x} \mu(n)$, where $\mu(n)$ is the Mobius 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 $2^{73}$.