# 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…
• Mathematics
75 Years of Mathematics of Computation
• 2020
It is shown that even though $\omega(n)$ and $\Omega(n), which respectively count the number of distinct and total prime factors of$n$have the same parity approximately 73.5\% of the time, these summatory functions exhibit quite different behaviors:$L(x)$is biased toward negative values, while$H(x%) is unbiased.
Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function $\vert M(x) \vert$ that $\left\vert M(x) \right\vert \sim \left[\frac{1}{\pi • Mathematics • 2020 We examine oscillations in a number of sums of arithmetic functions involving$\Omega(n)$, the total number of prime factors of$n$, and$\omega(n)$, the number of distinct prime factors of$n$. In • Materials Science Scientific Reports • 2021 Several novel formulas for the length of a Farey sequence of order n are proved and one of them is the most efficient algorithm for computing O(n) memory. The Mertens function, M(x) := ∑ n≤x μ(n), is defined as the summatory function of the Möbius function for x ≥ 1. The inverse function sequence {g(n)}n≥1 taken with respect to Dirichlet convolution is The Mertens function, M(x) ∶= ∑n≤x μ(n), is defined as the summatory function of the classical Möbius function for x ≥ 1. The inverse function g−1(n) ∶= (ω + 1)−1(n) taken with respect to Dirichlet The sieve of Eratosthenes will be able to use it to factor integers, and not just to produce lists of consecutive primes, and also has close ties to Voronoi's work on the Dirichlet divisor problem. Two elementary formulae for Mertens function$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
• Mathematics
• 2022
. 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.
• Mathematics
• 2021
Let ϝ(n) denote a multiplicative function with range {−1, 0, 1}, and let F (x) = ∑bxc n=1 ϝ(n). Then F (x)/ √ x = a √ x + b + E(x), where a and b are constants and E(x) is an error term that either