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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

  • [1] P. X. Gallagher and Julia H. Mueller, Primes and zeros in short intervals, J. Reine Angew. Math. 303/304 (1978), 205–220. MR 514680
  • [2] D. A. Goldston, On the function 𝑆(𝑇) in the theory of the Riemann zeta-function, J. Number Theory 27 (1987), no. 2, 149–177. MR 909834, 10.1016/0022-314X(87)90059-X
  • [3] D. A. Goldston, On the pair correlation conjecture for zeros of the Riemann zeta-function, J. Reine Angew. Math. 385 (1988), 24–40. MR 931214, 10.1515/crll.1988.385.24
  • [4] D. A. Goldston and S. M. Gonek, Mean values of $ \zeta '/\zeta \left( s \right)$ and primes in short intervals, (in preparation).
  • [5] Daniel A. Goldston and Hugh L. Montgomery, Pair correlation of zeros and primes in short intervals, Analytic number theory and Diophantine problems (Stillwater, OK, 1984), Progr. Math., vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 183–203. MR 1018376
  • [6] H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR 0337821
  • [7] Atle Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), no. 6, 87–105. MR 0012624

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11N05

Retrieve articles in all journals with MSC: 11N05

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society