# 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
• 2020
Let $R(n) = \sum_{a+b=n} \Lambda(a)\Lambda(b)$, where $\Lambda(\cdot)$ is the von Mangoldt function. The function $R(n)$ is often studied in connection with Goldbach's conjecture. On the Riemann
• 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$. In The Mertens function, M(x) := ∑ n≤x μ(n), is defined as the summatory function of the Möbius function for x ≥ 1. The inverse function sequence {g(n)}n≥1 taken with respect to Dirichlet convolution is • Materials Science Scientific reports • 2021 Several novel formulas for the length of a Farey sequence of order n are proved and one of them is the most efficient algorithm for computing O(n) memory. The sieve of Eratosthenes will be able to use it to factor integers, and not just to produce lists of consecutive primes, and also has close ties to Voronoi's work on the Dirichlet divisor problem. Two elementary formulae for Mertens function$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
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• 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.
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• 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

## References

SHOWING 1-10 OF 16 REFERENCES

• Mathematics
• 1984
The Mertens conjecture states that  M(x)  1, where M(x) = n ≤ x Σ μ(n) , and μ(n) is the Mo bius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of
• Mathematics
Exp. Math.
• 2004
Based on the extrema found in the range 104 ≤ x ≤ 101010, it is conjecture that q(x) = ω±,().
The summatory function of the Möbius function is denoted M(x). In this article we deduce conditional results concerning M(x) assuming the Riemann hypothesis and a conjecture of Gonek and Hejhal on
• Mathematics
• 2003
textabstractIn the past, the Mertens function M(x), i.e. the sum of the Moebius function ?(n) for 1 <= n <= x, has been computed for x <= 1013. We describe the results obtained by extending this
• Computer Science
Exp. Math.
• 1996
We describe an elementary method for computing isolated values of M(x) = Σ n≤x μ(n), where μ is the Mobius function. The complexity of the algorithm is O(x 2/3 (log log x)1/3) time and O(x 1/3(log
• Mathematics, Computer Science
• 2008
This paper presents a two-parametric family of recursive formulas for computing M(n), the Mobius function for positive integers n, and some identities and recursive formulas found.
• Computer Science, Mathematics
PLDI '94
• 1994
This paper presents code sequences for division by arbitrary nonzero integer constants and run-time invariants using integer multiplication using a two's complement architecture, and treats unsigned division, signed division, and division where the result is known a priori.
• Computer Science
ISSAC '09
• 2009
Thanks to the new orthogonalization strategy, H-LLL is the first LLL-type algorithm that admits a natural vectorial description, which leads to a complexity upper bound that is proportional to the progress performed on the basis (for fixed dimensions).
• Mathematics
• 1987
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects
• Mathematics
Math. Comput.
• 2015
This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens’ conjecture by showing that limsupx!1