Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-29T15:25:24.053Z Has data issue: false hasContentIssue false

Almost-primes in arithmetic progressions and short intervals

Published online by Cambridge University Press:  24 October 2008

D. R. Heath-Brown
Affiliation:
Trinity College, Cambridge

Extract

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1978

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

REFERENCES

(1)Chen, J.-R.On the distribution of almost primes in an interval. Sci. Sinica 18 (1975), 611627.Google Scholar
(2)Halberstam, H. and Richert, H.-E.Sieve methods (London: Academic Press, 1974).Google Scholar
(3)Hardy, G. H. and Wright, E. M.An introduction to the, theory of numbers (Oxford: Clarendon Press, 1962).Google Scholar
(4)Huxley, M. N.On the difference between consecutive primes. Invent. Math. 15 (1972), 164170.CrossRefGoogle Scholar
(5)Ingham, A. E.The distribution of prime numbers (Cambridge Mathematical Tract No. 30, 1932).Google Scholar
(6)Montgomery, H. L. and Vaughan, R. C.Hilbert's inequality. J. London Math. Soc. (2), 8 (1974), 7382.CrossRefGoogle Scholar
(7)Motohashi, Y.On almost primes in arithmetic progressions. J. Math. Soc. Japan 28 (1976), 363383.Google Scholar
(8)Motohashi, Y.On almost primes in arithmetic progressions: III. Proc. Japan Acad. 52 (1976), 116118.Google Scholar
(9)Selberg, A.On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid. 47 (1943), no. 6, 87105.Google Scholar
(10)Titchmarsh, E. C.A divisor problem. Rend. Circ. Mat. Palermo 54 (1930), 414429.CrossRefGoogle Scholar
(11)Titchmarsh, E. C.The theory of the Riemann Zeta-functian (Oxford, 1939).Google Scholar