Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Chebyshev type estimates for Beurling generalized prime numbers


Author: Wen-Bin Zhang
Journal: Proc. Amer. Math. Soc. 101 (1987), 205-212
MSC: Primary 11N80; Secondary 11N37
DOI: https://doi.org/10.1090/S0002-9939-1987-0902528-8
MathSciNet review: 902528
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a Beurling generalized prime system for which the distribution function $ N(x)$ of the integers satisfies

$\displaystyle \int_1^\infty {{x^{ - 1}}} \left\{ {\mathop {\sup }\limits_{x \leqslant y} \frac{{\left\vert {N(y) - Ay} \right\vert}} {y}} \right\}dx < \infty $

with constant $ A > 0$. We shall prove that the Chebyshev type estimates

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

hold for the system. This gives a partial proof of one of Diamond's conjectures.

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 0242778 (39:4105)
  • [2] A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I, Acta Math. 68 (1937), 225-291.
  • [3] H. G. Diamond, A set of generalized numbers showing Beurling's theorem to be sharp, Illinois J. Math. 14 (1970), 29-34. MR 0252335 (40:5556)
  • [4] -, Chebyshev estimates for Beurling generalized prime numbers, Proc. Amer. Math. Soc. 39 (1973), 503-508. MR 0314782 (47:3332)
  • [5] -, Chebyshev type estimates in prime number theory, Séminaire de Théorie des Nombres, Année 1974-1975 (Univ. Bodeaux I, Talence) exposé n$ ^\circ$ 24.
  • [6] R. S. Hall, Beurling generalized prime number system in which the Chebyshev inequalities fail, Proc. Amer. Math. Soc. 40 (1973), 79-82. MR 0318085 (47:6634)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11N80, 11N37

Retrieve articles in all journals with MSC: 11N80, 11N37


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0902528-8
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society