Functiones et Approximatio Commentarii Mathematici

A PNT equivalence for Beurling numbers

Harold G. Diamond and Wen-Bin Zhang

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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.

Article information

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

First available in Project Euclid: 25 June 2012

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Zentralblatt MATH identifier

Primary: 11N80: Generalized primes and integers

Beurling generalized numbers sharp Mertens formula Prime Number Theorem equivalent relations


Diamond, Harold G.; Zhang, Wen-Bin. A PNT equivalence for Beurling numbers. Funct. Approx. Comment. Math. 46 (2012), no. 2, 225--234. doi:10.7169/facm/2012.46.2.7.

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