Chebyshev bounds for Beurling numbers
Tom 160 / 2013
Acta Arithmetica 160 (2013), 143-157 MSC: Primary 11N80. DOI: 10.4064/aa160-2-4
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.