# 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} }

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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Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function $\vert M(x) \vert$ that $ \left\vert M(x) \right\vert \sim \left[\frac{1}{\pi…

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The Mertens function, M(x) := ∑ n≤x μ(n), is defined as the summatory function of the Möbius function for x ≥ 1. The inverse function sequence {g(n)}n≥1 taken with respect to Dirichlet convolution is…

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Two elementary formulae for Mertens function $M(n)$ are obtained. With these formulae, $M(n)$ can be calculated directly and simply, which can be easily implemented by computer. $M (1) \sim M (2…

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. We use Perron formula arguments to establish explicit versions of M ( x ) ≪ x , M ( x ) ≪ x log x exp( − c 1 √ log x ), and M ( x ) ≪ x exp( − c 2 √ log x ), where M ( x ) is the Mertens function.…

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