Properties of the Beurling generalized primes
2011 (English)In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 131, no 1, 45-58 p.Article in journal (Refereed) Published
In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that lim(x ->infinity) 1/Inx Pi(p <= x)(1 - p(-1))(-1) = Ae(gamma), where gamma is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit M = lim(x ->infinity) (Sigma(p <= x) p(-1) - In(Inx)) exists. We also show that this limit coincides with lim(alpha -> 0+)(Sigma(p) p(-1)(In p)(-alpha) - 1/alpha); for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small vertical bar N(x)-vertical bar can be made for a Beurling prime number system Q not equal P. where P is the rational primes. We prove that for each c > 0 there exists a Q such that vertical bar N(x) - [x]vertical bar < cInx and conjecture that this is the best possible bound.
Place, publisher, year, edition, pages
2011. Vol. 131, no 1, 45-58 p.
IdentifiersURN: urn:nbn:se:kth:diva-24337DOI: 10.1016/j.jnt.2010.06.014ISI: 000283401800004OAI: oai:DiVA.org:kth-24337DiVA: diva2:346650
FunderKnut and Alice Wallenberg Foundation
Updated from submitted to published. QC 201203272010-09-022010-09-022012-03-27Bibliographically approved