Optimality of Chebyshev bounds for Beurling generalized numbers (Q2847831)

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 NEWLINE\[NEWLINE \int_1^{\infty} x^{-2} |N(x)-Ax|\, dx < \infty NEWLINE\]NEWLINE and NEWLINE\[NEWLINE (N(x)-Ax)x^{-1}\log x =O(1) NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE (N(x)-Ax)x^{-1}\log x =O(f(x)) NEWLINE\]NEWLINE do not imply the same Chebyshev bound.