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Trigonometric sums over primes II

Published online by Cambridge University Press:  18 May 2009

Glyn Harman
Affiliation:
Royal Holloway College, Egham Surrey, Tw20 Oex
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We write e(x) for e2πix, ∥x∥ for the distance of x from the nearest integer and use AB to mean |A|<c |B|, where c is a positive constant depending at most on k and e. The letter p always denotes a prime number; P2 represents a number with precisely two prime factors. We continue the investigations started in [6] and will make many references to the analysis there. Here we prove the following theorems.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

REFERENCES

1.Baker, R. C., Small fractional parts of the sequence ank, Michigan Math. J. 28 (1981), 223228.CrossRefGoogle Scholar
2.Baker, R. C. and Harman, G., Diophantine approximation by prime numbers, J. London Math. Soc. (2) 25 (1982), 201215.CrossRefGoogle Scholar
3.Estermann, T., A new result in the additive prime number theory, Quart. J. Math. Oxford Ser. (2) 8 (1937), 3238.CrossRefGoogle Scholar
4.Fomenko, O. M., The distribution of numbers with a fixed number of different prime divisors, Studies in number theory 5, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 82 (1979), 158164, 168.Google Scholar
5.Graham, S. W., Diophantine approximation by almost primes, unpublished.Google Scholar
6.Harman, G., Trigonometric sums over primes I, Mathematika, 28 (1981), 249254.CrossRefGoogle Scholar
7.Hua, L. K., Additive theory of prime numbers, Amer. Math. Soc. Transl. 13 (1965).Google Scholar
8.Montgomery, H. L., The analytic principle of the large sieve, Bull. Amer. Math. Soc. 84 (1978), 547567.CrossRefGoogle Scholar
9.Vaughan, R. C., Sommes trigonometriques sur les nombres premiers, C.R. Acad. Sci. Paris Ser. A, 285 (1977), 981983.Google Scholar
10.Vaughan, R. C., An elementary method in prime number theory, Ada Arith. 37 (1980), 111115.CrossRefGoogle Scholar
11.Vinogradov, I. M., On the estimates of a trigonometric sum over primes, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 225248.Google Scholar
12.Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers (English translation by Davenport, A. and Roth, K. F. of 1947 Russian monograph, Wiley-Interscience, 1954).Google Scholar
13.Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers (Russian revised edition, Moscow, 1971).Google Scholar