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Optimality of Chebyshev bounds for Beurling generalized numbers. (English) Zbl 1315.11086
Let \(N(x)\) and \(\pi(x)\) denote the counting functions of integers, respectively primes in a Beurling generalized number system. It is known that the conditions \[ \int_1^{\infty} x^{-2} |N(x)-Ax|\, dx < \infty \] and \[ (N(x)-Ax)x^{-1}\log x =O(1) \] imply the Chebyshev bound \(\pi(x)\ll x/\log x\). The authors show that given any positive valued function \(f\) satisfying \(\lim_{x\to \infty} f(x)=\infty\), the above first condition and \[ (N(x)-Ax)x^{-1}\log x =O(f(x)) \] do not imply the same Chebyshev bound.

MSC:
11N80 Generalized primes and integers
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