Optimality of Chebyshev bounds for Beurling generalized numbers
Volume 160 / 2013
Acta Arithmetica 160 (2013), 259-275
MSC: Primary 11N80.
DOI: 10.4064/aa160-3-3
Abstract
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 to satisfy the Chebyshev bound $\pi (x)\ll x/\log x$. Let $f(x)$ increase to infinity arbitrarily slowly. We give a construction showing that $\int _1^\infty |N(x)-Ax|x^{-2}\,dx<\infty $ and $x^{-1}(\log x) (N(x)-Ax)=O(f(x))$ do not imply the Chebyshev bound.
