An omega-result for Beurling generalized integers

We consider Beurling number systems with very well-behaved primes, in the sense that $\psi (x) = x + O(x^{\alpha })$ for some $\alpha \lt 1/2$. We investigate how small the error term in the asymptotic formula for the integer-counting function $N(x)$ can be for such systems. In particular, we show that \[ N(x) - \rho x = \Omega (\sqrt{x}\,\mathrm e^{-(\log x)^{\beta }}) \] for any $\beta \gt 2/3$.