# zbMATH — the first resource for mathematics

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
Full Text: