$M(x)=o(x)$ Estimates for Beurling numbers

TL;DR

This paper investigates conditions under which the Beurling number system satisfies the $M(x)=o(x)$ estimate, extending classical prime number theory results to generalized number systems.

Contribution

The paper introduces two conditions that guarantee the Beurling analogue of the $M(x)=o(x)$ estimate and provides examples showing failures without these conditions.

Findings

Two conditions ensure $M(x)=o(x)$ for Beurling numbers

Examples demonstrate failure of the estimate when conditions are not met

Extends classical prime number results to generalized number systems

Abstract

In classical prime number theory there are several asymptotic formulas said to be "equivalent" to the PNT. One is the bound $M(x) = o(x)$ for the sum function of the Moebius function. For Beurling generalized numbers, this estimate is not an unconditional consequence of the PNT. Here we give two conditions that yield the Beurling version of the $M(x)$ bound, and examples illustrating failures when these conditions are not satisfied.