Properties of the Beurling generalized primes |
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Authors: | Rikard Olofsson |
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Affiliation: | Department of Mathematics, Uppsala University, P.O. Box 480, SE-75106 Uppsala, Sweden |
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Abstract: | TextIn 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.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Kw3iNo3fAbk/. |
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Keywords: | Analytic number theory Zeta functions Beurling primes Mertens' theorem Beurling's conjecture |
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