A note on the number of primes in short intervals
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- by D. A. Goldston and S. M. Gonek PDF
- Proc. Amer. Math. Soc. 108 (1990), 613-620 Request permission
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.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 613-620
- MSC: Primary 11N05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1002158-6
- MathSciNet review: 1002158