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

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 $\frac{1}{2}$.