Chebyshev type estimates for Beurling generalized prime numbers

Abstract

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