Optimality of Chebyshev bounds for Beurling generalized numbers

Harold G. Diamond¹ ; Wen-Bin Zhang²

¹ Department of Mathematics University of Illinois Urbana, IL 61801, U.S.A.
² 920 West Lawrence Ave. #1112 Chicago, IL 60640, U.S.A.

Acta Arithmetica, Tome 160 (2013) no. 3, pp. 259-275.

Résumé

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.

Détail

DOI : 10.4064/aa160-3-3

Keywords: counting function integers beurling generalized number system satisfies int infty ax infty log ax counting function primes system known satisfy chebyshev bound log increase infinity arbitrarily slowly construction showing int infty ax infty log ax imply chebyshev bound