# Randomness of Möbius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis

@article{Mussardo2021RandomnessOM, title={Randomness of M{\"o}bius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis}, author={Giuseppe Mussardo and Andr{\'e} LeClair}, journal={Journal of Statistical Mechanics: Theory and Experiment}, year={2021}, volume={2021} }

The validity of the Riemann hypothesis (RH) on the location of the non-trivial zeros of the Riemann ζ-function is directly related to the growth of the Mertens function M(x)=∑k=1xμ(k) , where μ(k) is the Möbius coefficient of the integer k; the RH is indeed true if the Mertens function goes asymptotically as M(x) ∼ x 1/2+ϵ , where ϵ is an arbitrary strictly positive quantity. We argue that this behavior can be established on the basis of a new probabilistic approach based on the global…

## One Citation

### Riemann Zeroes from a Parametric Oscillator analyzed with Adiabatic Invariance, Hill Equation and the Least Action Principle

- Physics
- 2022

. Adiabatic Invariance (AdI), Hill´s Equation formalism (HEF), and the Least Action Principle (LAP), three relevant tools of theoretical physics are here separately applied to a one- dimensional…

## References

SHOWING 1-10 OF 140 REFERENCES

### Generalized Riemann hypothesis and stochastic time series

- MathematicsJournal of Statistical Mechanics: Theory and Experiment
- 2018

Using the Dirichlet theorem on the equidistribution of residue classes modulo q and the Lemke Oliver–Soundararajan conjecture on the distribution of pairs of residues on consecutive primes, we show…

### Riemann Hypothesis and Random Walks: The Zeta Case

- MathematicsSymmetry
- 2021

A new algorithm for computing very high Riemann zeros, and the googol-th zero, namely 10100-thZero to over 100 digits, is described, far beyond what is currently known.

### Some Riemann Hypotheses from random walks over primes

- MathematicsCommunications in Contemporary Mathematics
- 2018

The aim of this paper is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of [Formula: see text]-functions,…

### On the Order of the Mertens Function

- MathematicsExp. Math.
- 2004

Based on the extrema found in the range 104 ≤ x ≤ 101010, it is conjecture that q(x) = ω±,().

### The Distribution of the Summatory Function of the Möbius Function

- Mathematics
- 2003

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…

### Random Matrix Theory and ζ(1/2+it)

- Mathematics
- 2000

Abstract: We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the…

### Generalized Riemann hypothesis, time series and normal distributions

- MathematicsJournal of Statistical Mechanics: Theory and Experiment
- 2019

L functions based on Dirichlet characters are natural generalizations of the Riemann function: they both have series representations and satisfy an Euler product representation, i.e. an infinite…

### A pseudo zeta function and the distribution of primes.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 2000

It may be shown that the pseudo zeta function C(s) has no complex zeros whatsoever, which means that the complexZeta function are associated with the irregularity of the distribution of the primes.

### The Riemann-Siegel expansion for the zeta function: high orders and remainders

- MathematicsProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
- 1995

On the critical line s ═ ½ + it (t real), Riemann’s zeta function can be calculated with high accuracy by the Riemann-Siegel expansion. This is derived here by elementary formal manipulations of the…

### The Riemann Zeros and Eigenvalue Asymptotics

- MathematicsSIAM Rev.
- 1999

It is speculated that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl=XP, and very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues.