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})$,forsome$\rho >0$and$\alpha, \beta, \gamma<1$\mathrm{.}Weshowthatthetwolargestof$\alpha,\beta,\gamma$mustbeequalandatleast$\frac{1}{2}$.