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 < 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 = ohm(root x e-(log x)beta ) for any beta > 2/3.