Chebyshev type estimates for Beurling generalized prime numbers. II

Abstract:

Let $N(x)$ be the distribution function of the integers in a Beurling generalized prime system. The Chebyshev type estimates for Beurling generalized prime numbers in the general case \[ N(x) = x\sum \limits _{\nu = 1}^n {{A_\nu }} {\log ^{{\rho _\nu } - 1}}x + O(x{\log ^{ - \gamma }}x)\] is a long standing question. In this paper we shall give an affirmative answer to the question by proving 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 even under weaker condition \[ \int _1^\infty {{x^{ - 1}}} \left \{ {\sup \limits _{x < \infty } {y^{ - 1}}\left | {N(y) - y\sum \limits _{\nu = 1}^n {{A_\nu }} {{\log }^{{\rho _\nu } - 1}}y} \right |} \right \} dx < \infty \] with $\rho _n=\tau \geq 1$, $0<\rho _1<\rho _2 <\cdots < \rho _n$, and $A_n > 0$. This generalizes a result of Diamond and a result of the present author.