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), 613620
MSC:
Primary 11N05
MathSciNet review:
1002158
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let , where the sum is over prime powers. H. L. Montgomery has shown that on the Riemann hypothesis, there is a positive constant such that for each , provided that is sufficiently large. Here we prove a slightly stronger result from which we deduce a lower bound of the same order.
 [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
(80b:10060)
 [2]
D.
A. Goldston, On the function 𝑆(𝑇) in the theory of
the Riemann zetafunction, J. Number Theory 27
(1987), no. 2, 149–177. MR 909834
(89a:11086), http://dx.doi.org/10.1016/0022314X(87)90059X
 [3]
D.
A. Goldston, On the pair correlation conjecture for zeros of the
Riemann zetafunction, J. Reine Angew. Math. 385
(1988), 24–40. MR 931214
(89c:11133), http://dx.doi.org/10.1515/crll.1988.385.24
 [4]
D. A. Goldston and S. M. Gonek, Mean values of 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
(90h:11084)
 [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
(49 #2590)
 [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
(7,48e)
 [1]
 P. X. Gallagher and J. Mueller, Primes and zeros in short intervals, J. Reine Angew. Math. 303/304 (1978), 205220. MR 514680 (80b:10060)
 [2]
 D. A. Goldston, On the function in the theory of the Riemann zetafunction, J. Number Theory 27 (1987), 149177. MR 909834 (89a:11086)
 [3]
 , On the pair correlation conjecture for zeros of the Riemann zetafunction, J. Reine Angew. Math. 385 (1988), 2440. MR 931214 (89c:11133)
 [4]
 D. A. Goldston and S. M. Gonek, Mean values of and primes in short intervals, (in preparation).
 [5]
 D. A. Goldston and H. L. Montgomery, Pair correlation of zeros and primes in short intervals, in Analytic Number Theory and Diophantine Problems, Proc. of a conference at Oklahoma State University, 1984, Birkhäuser, BostonBaselStuttgart 1987, 183203. MR 1018376 (90h:11084)
 [6]
 H. L. Montgomery, The pair correlation of zeros of the zeta function, Proc. Symp. Pure Math. 24 (1973), 181193. MR 0337821 (49:2590)
 [7]
 A. Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), 87105. MR 0012624 (7:48e)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
11N05
Retrieve articles in all journals
with MSC:
11N05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010021586
PII:
S 00029939(1990)10021586
Article copyright:
© Copyright 1990
American Mathematical Society
