$M(x)=o(x)$ Estimates for Beurling numbers
  In classical prime number theory there are several asymptotic formulas said to be "equivalent" to the PNT. One is the bound $M(x) = o(x)$ for the sum function of the Moebius function. For Beurling generalized numbers, this estimate is not an unconditional consequence of the PNT. Here we give two conditions that yield the Beurling version of the $M(x)$ bound, and examples illustrating failures when these conditions are not satisfied. 
Gregory Debruyne, Harold G. Diamond and Jasson Vindas
math.NT