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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Chebyshev estimates for Beurling generalized prime numbers

Author: Harold G. Diamond
Journal: Proc. Amer. Math. Soc. 39 (1973), 503-508
MSC: Primary 10H20
MathSciNet review: 0314782
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Abstract: We consider a Beurling generalized number system for which the counting function of the integers satisfies

$\displaystyle N(x) = cx + O(x{\log ^{ - \gamma }}x)$

for some positive $ c$ and $ \gamma $. Beurling showed that the prime number theorem must hold if $ \gamma > \tfrac{3}{2}$, but it can fail to hold if $ \gamma \leqq \tfrac{3}{2}$. It was shown by Hall in the preceding article that the Chebyshev prime counting estimates

$\displaystyle 0 < \mathop {\lim \inf }\limits_{x \to \infty } \frac{{\psi (x)}}... ...quad \mathop {\lim \sup }\limits_{x \to \infty } \frac{{\psi (x)}}{x} < \infty $

can fail if $ \gamma < 1$. Here we shall prove that these estimates hold for any system satisfying $ \gamma > 1$. The proof uses a convolution approximate inverse of the measure $ dN$.

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

  • [1] Paul T. Bateman and Harold G. Diamond, Asymptotic distribution of Beurling’s generalized prime numbers, Studies in Number Theory, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), 1969, pp. 152–210. 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] T. M. K. Davison, A Tauberian theorem and analogues of the prime number theorem, Canad. J. Math. 20 (1968), 362–367. MR 0224569 (37 #168)
  • [4] Harold G. Diamond, Asymptotic distribution of Beurling’s generalized integers, Illinois J. Math. 14 (1970), 12–28. MR 0252334 (40 #5555)
  • [5] R. S. Hall, Theorems about Beurling's generalized primes and the associated zeta function, Ph.D. Thesis, University of Illinois, Urbana, Ill., 1967.
  • [6] S. L. Segal, A Tauberian theorem for Dirichlet convolutions, Illinois J. Math. 13 (1969), 316–320. MR 0238779 (39 #143)

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PII: S 0002-9939(1973)0314782-4
Article copyright: © Copyright 1973 American Mathematical Society

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