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Chebyshev bounds for Beurling numbers. (English) Zbl 1309.11069
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.

MSC:
11N80 Generalized primes and integers
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