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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 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
MathSciNet review: 902528
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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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0902528-8
PII: S 0002-9939(1987)0902528-8
Article copyright: © Copyright 1987 American Mathematical Society