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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Chebyshev type estimates for Beurling generalized prime numbers. II


Author: Wen-Bin Zhang
Journal: Trans. Amer. Math. Soc. 337 (1993), 651-675
MSC: Primary 11N80; Secondary 11N37
MathSciNet review: 1112550
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Abstract: Let $ N(x)$ be the distribution function of the integers in a Beurling generalized prime system. The Chebyshev type estimates for Beurling generalized prime numbers in the general case

$\displaystyle N(x) = x\sum\limits_{\nu = 1}^n {{A_\nu }} {\log ^{{\rho _\nu } - 1}}x + O(x{\log ^{ - \gamma }}x)$

is a long standing question. In this paper we shall give an affirmative answer to the question by proving that the Chebyshev type 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 $

hold even under weaker condition

$\displaystyle \int_1^\infty {{x^{ - 1}}} \left\{ {\mathop {\sup }\limits_{x < \... ...n {{A_\nu }} {{\log }^{{\rho _\nu } - 1}}y} \right\vert} \right\}\,dx < \infty $

with $ \rho_n=\tau \geq 1$, $ 0<\rho_1<\rho_2 <\cdots < \rho_n$, and $ A_n > 0$. This generalizes a result of Diamond and a result of the present author.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1112550-4
PII: S 0002-9947(1993)1112550-4
Keywords: Beurling generalized prime numbers, Chebyshev type estimates, approximate convolution inverse
Article copyright: © Copyright 1993 American Mathematical Society