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Beurling generalized prime number systems in which the Chebyshev inequalities fail


Author: R. S. Hall
Journal: Proc. Amer. Math. Soc. 40 (1973), 79-82
MSC: Primary 10H40
DOI: https://doi.org/10.1090/S0002-9939-1973-0318085-3
MathSciNet review: 0318085
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Abstract: It is proved that there exist systems of generalized primes in which the asymptotic distribution of integers is $ N(x) = Ax + O(x \cdot {\log ^{ - \gamma }}x)$ with $ A > 0$ and $ \gamma \in [0,1)$ and in which the Chebyshev inequalities

$\displaystyle \mathop {\lim \inf }\limits_{x \to \infty } \frac{{\pi (x)\log x}... ...\mathop {\lim \sup }\limits_{x \to \infty } \frac{{\pi (x)\log x}}{x} < \infty $

do not hold.

References [Enhancements On Off] (What's this?)

  • [1] P. T. Bateman and H. G. Diamond, Asymptotic distribution of Beurling's generalized prime numbers, Studies in Number Theory, vol. 6, Math. Assoc. Amer., Prentice-Hall, Englewood Cliffs, N.J., 1969, pp. 152-210. MR 39 #4105. MR 0242778 (39:4105)
  • [2] A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I, Acta Math. 68 (1937), 255-291.
  • [3] R. S. Hall, Theorems about Beurling's generalized primes and the associated zeta function, Ph.D. Thesis, University of Illinois, Urbana, Ill., 1967.

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DOI: https://doi.org/10.1090/S0002-9939-1973-0318085-3
Article copyright: © Copyright 1973 American Mathematical Society

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