Optimality of Chebyshev bounds for Beurling generalized numbers
Harold George Diamond, Wenbin Zhang · Acta Arithmetica · 2013
If the counting function $N(x)$ of integers of a Beurling generalized number system satisfies both $\int _1^\infty x^{-2}|N(x)-Ax|\,dx<\infty $ and $x^{-1}(\log x) (N(x)-Ax)=O(1)$, then the counting function $\pi (x)$ of the primes of this system is known