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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.

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

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  • [2] A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I, Acta Math. 68 (1937), 225-291.
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