Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T18:12:37.718Z Has data issue: false hasContentIssue false
  • English
  • Français

PRIME NUMBER THEOREM EQUIVALENCES AND NON-EQUIVALENCES

Part of: Multiplicative number theory

Published online by Cambridge University Press:  29 November 2017

Harold G. Diamond  and
Wen-Bin Zhang
Harold G. Diamond
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, U.S.A. email [email protected]
Wen-Bin Zhang
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, U.S.A. email [email protected]
Get access

Abstract

There are several formulas in classical prime number theory that are said to be “equivalent” to the Prime Number Theorem. For Beurling generalized numbers, not all such implications hold unconditionally. Here we investigate conditions under which the Beurling version of these relations do or do not hold.

MSC classification

Primary: 11N80: Generalized primes and integers
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 852 - 862
Copyright
Copyright © University College London 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bateman, P. T. and Diamond, H. G., Asymptotic distribution of Beurling’s generalized prime numbers. In Studies in Number Theory (Mathematical Association of America, Studies in Mathematics 6 ) (ed. LeVeque, W. J.), Prentice-Hall (Englewood Cliffs, NJ, 1969), 152–210.Google Scholar
Bateman, P. T. and Diamond, H. G., Analytic Number Theory: An Introductory Course (Monographs in Number Theory 1 ), World Scientific (Singapore, 2004). Reprinted with minor changes 2009.CrossRefGoogle Scholar
DeBruyne, G., Diamond, H. G. and Vindas, J., M (x) = o (x) estimates for Beurling numbers. J. Théor. Nombres Bordeaux (to appear).Google Scholar
Diamond, H. G. and Zhang, W. B., A PNT equivalence for Beurling numbers. Funct. Approx. Comment. Math. 46 2012, 225–234.Google Scholar
Diamond, H. G. and Zhang, W. B., Beurling Generalized Numbers (Mathematical Surveys and Monographs 213 ), American Mathematical Society (Providence, RI, 2016).Google Scholar
Hardy, G. H., Divergent Series, Clarendon Press (Oxford, 1949).Google Scholar
Montgomery, H. L. and Vaughan, R. C., Multiplicative Number Theory: I. Classical Theory (Cambridge Studies in Advanced Mathematics 97 ), Cambridge University Press (Cambridge, 2007).Google Scholar