Chebyshev bounds for Beurling numbers

Volume 160 / 2013

Harold G. Diamond, Wen-Bin Zhang Acta Arithmetica 160 (2013), 143-157 MSC: Primary 11N80. DOI: 10.4064/aa160-2-4

Abstract

The first author conjectured that Chebyshev-type prime bounds hold for Beurling generalized numbers provided that the counting function $N(x)$ of the generalized integers satisfies the $L^1$ condition \[ \int _1^\infty |N(x) - Ax|\,dx/x^2 < \infty \] for some positive constant $A$. This conjecture was shown false by an example of Kahane. Here we establish the Chebyshev bounds using the $L^1$ hypothesis and a second integral condition.

Authors

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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image