The average order of the Möbius function for Beurling primes

Abstract

In this paper, we study the counting functions $ψ_\mathcal{P}(x)$, $N_\mathcal{P}(x)$ and $M_\mathcal{P}(x)$ of a generalized prime system $\mathcal{N}$. Here $M_\mathcal{P}(x)$ is the partial sum of the Möbius function over $\mathcal{N}$ not exceeding $x$. In particular, we study these when they are asymptotically well-behaved, in the sense that $ψ_{\cal{P}}(x) = x+O({x^{ α+ε}})$, $N_{\cal{P}}(x) = ρx+O({x^{ β+ε}})$ and $ M_\mathcal{P}(x) = O(x^{γ+ε})$, for some $ρ>0$ and $α, β, γ<1$. We show that the two largest of $α,β,γ$ must be equal and at least $\frac{1}{2}$.