Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T05:16:23.242Z Has data issue: false hasContentIssue false

On Bombieri and Davenport's theorem concerning small gaps between primes

Published online by Cambridge University Press:  26 February 2010

D. A. Goldston
Affiliation:
Department of Mathematics and Computer Science, San Jose State University, San Jose, CA 95192, USA.
Get access

Extract

§1. Introduction. In this paper we give a new proof of a theorem of Bombieri and Davenport [2, Theorem 1]. Let t(–k) = t(k) be real,

where e(u) = e2πiu. Let p and p' denote primes, k an integer, and define

MSC classification

Type
Research Article
Copyright
Copyright © University College London 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bombieri, E.. On twin almost primes. Acta Aritk., 28 (1975), 177193; Corrigendum,CrossRefGoogle Scholar
Bombieri, E.. On twin almost primes. Acta Aritk., 28 (1976), 457461.CrossRefGoogle Scholar
2.Bombieri, E. and Davenport, H.. Small differences between prime numbers. Proc. Roy. Soc. A, 293 (1966), 118.Google Scholar
3.Bombieri, E., Friedlander, J. B. and Iwaniec, H.. Primes in arithmetic progressions to large moduli. III. J. of the American Math. Soc., 2 (1989), 215224.CrossRefGoogle Scholar
4.Goldston, D. A.. Linnik's theorem on Goldbach numbers in short intervals. Glasgow Math. J., 32 (1990), 285297.CrossRefGoogle Scholar
5.Goldston, D. A.. On Hardy and Littlewood's Contribution to the Goldbach Conjecture. Submitted, 1989 Amalfi Conference proceedings.Google Scholar
6.Graham, S.. An asymptotic estimate related to Selberg's sieve. J. of Number Theory, 10 (1978), 8394.CrossRefGoogle Scholar
7.Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers, 4th ed. (Oxford Univ. Press, Oxford, 1960).Google Scholar
8.Heath-Brown, D. R.. The ternary Goldbach problem. Revista Matemática Iberoamericana, 1 (1985), 4559.CrossRefGoogle Scholar
9.Huxley, M. N.. Irregularity in shifted sequences. J. of Number Theory, 4 (1972), 437454.CrossRefGoogle Scholar
10.Huxley, M. N.. Small differences between prime numbers II. Mathematika, 24 (1977), 142152.CrossRefGoogle Scholar
11.Maier, Helmut. Small differences between prime numbers. Michigan Math. J., 35 (1988), 323344.CrossRefGoogle Scholar
12.Turán, P.. On the twin prime problem I. Publ. Math. Inst. Hung. Acad. Sci. Ser. 4. 9 (1964), 247261.Google Scholar