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

 

A note on the number of primes in short intervals


Authors: D. A. Goldston and S. M. Gonek
Journal: Proc. Amer. Math. Soc. 108 (1990), 613-620
MSC: Primary 11N05
MathSciNet review: 1002158
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Abstract: Let $ J\left( {\beta ,T} \right) = \int_1^{{T^\beta }} {{{\left( {\sum\nolimits_{x < {p^k} \leq x + x/T} {\log p - x/T} } \right)}^2}dx/{x^2}} $, where the sum is over prime powers. H. L. Montgomery has shown that on the Riemann hypothesis, there is a positive constant $ {C_0}$ such that for each $ \beta \geq 1,J\left( {\beta ,T} \right) \leq {C_0}\beta {\log ^2}T/T$, provided that $ T$ is sufficiently large. Here we prove a slightly stronger result from which we deduce a lower bound of the same order.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1002158-6
PII: S 0002-9939(1990)1002158-6
Article copyright: © Copyright 1990 American Mathematical Society