# The average order of the M\"{o}bius function for Beurling primes

## TL;DR

This paper investigates the average order of the Möbius function within generalized prime systems, establishing bounds and relationships among key asymptotic parameters under certain regularity conditions.

## Contribution

It introduces bounds on the asymptotic behavior of counting functions for generalized primes and relates the largest exponents governing their growth.

## Key findings

- The maximum of the exponents α, β, γ is at least 1/2.
- The two largest exponents among α, β, γ are equal.
- Provides conditions under which these asymptotic estimates hold.

## Abstract

In this paper, we study the counting functions $\psi_\mathcal{P}(x)$, $N_\mathcal{P}(x)$ and $M_\mathcal{P}(x)$ of a generalized prime system $\mathcal{N}$. Here $M_\mathcal{P}(x)$ is the partial sum of the M\"{o}bius function over $\mathcal{N}$ not exceeding $x$. In particular, we study these when they are asymptotically well-behaved, in the sense that $\psi_{\cal{P}}(x) = x+O({x^{ \alpha+\epsilon }})$, $N_{\cal{P}}(x) = \rho x+O({x^{ \beta+\epsilon }})$ and $ M_\mathcal{P}(x) = O(x^{\gamma+\epsilon})$, for some $\rho >0$ and $\alpha, \beta, \gamma<1$. We show that the two largest of $\alpha,\beta,\gamma$ must be equal and at least $\frac{1}{2}$.