On the estimate $M(x)=o(x)$ for Beurling generalized numbers

TL;DR

This paper proves that in Beurling number systems, the M"{o}bius function sum satisfies $M(x)=o(x)$ under certain prime distribution conditions, extending classical results to generalized number systems.

Contribution

It establishes the asymptotic bound $M(x)=o(x)$ for Beurling numbers when the prime distribution is a monotone perturbation of classical primes or the logarithmic integral.

Findings

$M(x)=o(x)$ holds under specified prime distribution conditions

Results extend classical prime number theorem to Beurling systems

Applicable to systems with monotone prime perturbations

Abstract

We show that the sum function of the M\"{o}bius function of a Beurling number system must satisfy the asymptotic bound $M(x)=o(x)$ if it satisfies the prime number theorem and its prime distribution function arises from a monotone perturbation of either the classical prime numbers or the logarithmic integral.