Geodesic
Chebyshev bounds for Beurling numbers
Harold G. Diamond 1   ; Wen-Bin Zhang 2
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. 2, pp. 143-157 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/aa160-2-4
Keywords: first author conjectured chebyshev type prime bounds beurling generalized numbers provided counting function generalized integers satisfies condition int infty infty positive constant conjecture shown false example kahane here establish chebyshev bounds using hypothesis second integral condition

Harold G. Diamond  1   ; Wen-Bin Zhang  2

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. 
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Harold G. Diamond; Wen-Bin Zhang. Chebyshev bounds for Beurling numbers. Acta Arithmetica, Tome 160 (2013) no. 2, pp. 143-157. doi: 10.4064/aa160-2-4

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