On the estimate \(M(x)=o(x)\) for Beurling generalized numbers

We show that the sum function of the Möbius function of a Beurling number system must satisfy the asymptotic bound \(M(x)=o(x)\) $M(x)=o(x)$ if it satisfies the prime number theorem and its prime distribution function arises from a monotone perturbation of either the classical prime numbers or the logarithmic integral.