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… 

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