The average order of the Möbius function for Beurling primes

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 Möbius function over $\mathcal{N}$ not exceeding $x$. In particular, we study these when they are asymptotically well-behaved, in the sense that ${\psi }_{\mathcal{P}}(x)=x+O(x^{\alpha +\epsilon })$, $N_{\mathcal{P}}(x)=\rho x+O(x^{\beta +\epsilon })$ 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 $\frac{1}{2}$.

6 pages