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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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