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Well-behaved Beurling primes and integers. (English) Zbl 1154.11335

Summary: We study generalized prime systems for which both the prime and integer counting functions are asymptotically well-behaved, in the sense that they are approximately \(li(x)\) and \(\rho x\), respectively (where \(\rho\) is a positive constant), with error terms of order \(O(x^{\theta_1})\) and \(O(x^{\theta_2})\) for some \(\theta_1, \theta_2 < 1\). We show that it is impossible to have both \(\theta_1\) and \(\theta_2\) less than \(\frac{1}{2}\).

MSC:

11N80 Generalized primes and integers
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