Chebyshev type estimates for Beurling generalized prime numbers

Abstract:

We consider a Beurling generalized prime system for which the distribution function $N(x)$ of the integers satisfies \[ \int _1^\infty {{x^{ - 1}}} \left \{ {\sup \limits _{x \leqslant y} \frac {{\left | {N(y) - Ay} \right |}} {y}} \right \}dx < \infty \] with constant $A > 0$. We shall prove that the Chebyshev type estimates \[ 0 < \lim \inf \limits _{x \to \infty } \frac {{\psi (x)}} {x},\quad \lim \sup \limits _{x \to \infty } \frac {{\psi (x)}} {x} < \infty \] hold for the system. This gives a partial proof of one of Diamond’s conjectures.