Properties of the Beurling generalized primes

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• 作者：Rikard Olofsson
• 关键词：Analytic number theory ; Zeta functions ; Beurling primes ; Mertens' theorem ; Beurling's conjecture
• 刊名：Journal of Number Theory
• 出版年：2011
• 出版时间：January 2011
• 年：2011
• 卷：131
• 期：1
• 页码：45-58
• 全文大小：183 K

文摘

Text

In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that , where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small N(x)−[x] can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that and conjecture that this is the best possible bound.

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