A PNT equivalence for Beurling numbers

A PNT equivalence for Beurling numbers

Harold G. Diamond and Wen-Bin Zhang

Source: Funct. Approx. Comment. Math. Volume 46, Number 2 (2012), 225-234.

Abstract

In classical prime number theory, several relations are considered to be equivalent to the Prime Number Theorem. For Beurling generalized numbers, some auxiliary conditions may be needed to deduce one relation from another one. We show conditions under which the Beurling analog of the sharp version of Mertens' sum formula does or does not hold.

First Page: Show

Primary Subjects: 11N80

Keywords: Beurling generalized numbers; sharp Mertens formula; Prime Number Theorem; equivalent relations

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.facm/1340628404
Digital Object Identifier: doi:10.7169/facm/2012.46.2.7
Zentralblatt MATH identifier: 06074841
Mathematical Reviews number (MathSciNet): MR2931668

back to Table of Contents

References

P. T. Bateman and H. G. Diamond, Asymptotic distribution of Beurling's generalized prime numbers, in Studies in Number Theory (W. J. LeVeque, ed.), Math. Assn. of America, 1969, pp. 152–210. MR0242778 (39 #4105).

Mathematical Reviews (MathSciNet): MR242778

Zentralblatt MATH: 0216.31403

ibid, Analytic Number Theory: An Introductory Course, World Scientific Pub. Co., 2004. MR2111739 (2005h:11208).

Mathematical Reviews (MathSciNet): MR2111739

A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I, Acta Math. 68 (1937), 255–291.

H. G. Diamond, Asymptotic distribution of Beurling's generalized integers, Illinois J. Math. 14 (1970), pp. 12–28. MR0252334 (40 #5555).

Mathematical Reviews (MathSciNet): MR252334

Zentralblatt MATH: 0186.36403

Project Euclid: euclid.ijm/1256053295

H. L. Montgomery and R. C. Vaughan, Multiplicative number theory, I. Classical theory, Cambridge Studies in Adv. Math. 97, Cambridge Univ. Press, 2007. MR2378655 (2009b:11001).

Mathematical Reviews (MathSciNet): MR2378655

Zentralblatt MATH: 1142.11001

previous :: next

Functiones et Approximatio Commentarii Mathematici