Diamond, Harold G.; Zhang, Wen-Bin Chebyshev bounds for Beurling numbers. (English) Zbl 1309.11069 Acta Arith. 160, No. 2, 143-157 (2013). Summary: The first author [Proc. Am. Math. Soc. 39, 503–508 (1973; Zbl 0268.10036)] 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. Cited in 3 Documents MSC: 11N80 Generalized primes and integers Keywords:Beurling generalized numbers; Chebyshev prime bounds; Fejér kernel estimates; Wiener theorems Citations:Zbl 0268.10036 PDFBibTeX XMLCite \textit{H. G. Diamond} and \textit{W.-B. Zhang}, Acta Arith. 160, No. 2, 143--157 (2013; Zbl 1309.11069) Full Text: DOI