Chebyshev estimates for Beurling generalized prime numbers

Abstract:

We consider a Beurling generalized number system for which the counting function of the integers satisfies \[ N(x) = cx + O(x{\log ^{ - \gamma }}x)\] for some positive $c$ and $\gamma$. Beurling showed that the prime number theorem must hold if $\gamma > \tfrac {3}{2}$, but it can fail to hold if $\gamma \leqq \tfrac {3}{2}$. It was shown by Hall in the preceding article that the Chebyshev prime counting estimates \[ 0 < \lim \inf \limits _{x \to \infty } \frac {{\psi (x)}}{x};\quad \lim \sup \limits _{x \to \infty } \frac {{\psi (x)}}{x} < \infty \]can fail if $\gamma < 1$. Here we shall prove that these estimates hold for any system satisfying $\gamma > 1$. The proof uses a convolution approximate inverse of the measure $dN$.