Optimality of Chebyshev bounds for Beurling generalized numbers

Volume 160 / 2013

Harold G. Diamond, Wen-Bin Zhang Acta Arithmetica 160 (2013), 259-275 MSC: Primary 11N80. DOI: 10.4064/aa160-3-3


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.


  • Harold G. DiamondDepartment of Mathematics
    University of Illinois
    Urbana, IL 61801, U.S.A.
  • Wen-Bin Zhang920 West Lawrence Ave. #1112
    Chicago, IL 60640, U.S.A.

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