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

DOI:
https://doi.org/10.1090/S0002-9939-1990-1002158-6

MathSciNet review:
1002158

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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****[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**, https://doi.org/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**, https://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****[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**

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DOI:
https://doi.org/10.1090/S0002-9939-1990-1002158-6

Article copyright:
© Copyright 1990
American Mathematical Society