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Trigonometric sums over primes I

Published online by Cambridge University Press:  26 February 2010

Glyn Harman
Glyn Harman
Affiliation:
Royal Holloway College, Egham, Surrey TW20 OEX.
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Extract

We write e(x) for e2πix and let ‖x‖ denote the distance of x from the nearest integer. The notation A ≪ B will mean |A| ≤ C|B| where C is a positive constant depending at most on an arbitrary positive number ε, and on an integer k. The letter p always denotes a prime number. The main results of the present paper are as follows.

Type
Research Article
Information
Mathematika , Volume 28 , Issue 2 , December 1981 , pp. 249 - 254
Copyright
Copyright © University College London 1981

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References

1.Baker, R. C. and Harman, G.. “Small fractional parts of quadratic forms”, to appear in Proc. Edinburgh Math. Soc.Google Scholar
2.Baker, R. C. and Harman, G.. “Diophantine approximation by prime numbers”, to appear in J. London Math. Soc.Google Scholar
3.Jing-Run, Chen. “Estimates for trigonometric sums”, Chinese Mathematics, 6 (1965), 163–167.Google Scholar
4.Ghosh, A.. “The distribution of αp2 modulo one”, Proc. London Math. Soc. (3), 42 (1981), 252–269.CrossRefGoogle Scholar
5.Harman, G., “Trigonometric sums over primes II”, to appear.Google Scholar
6.Schmidt, W. M., “Small fractional parts of polynomials”, Regional Conference Series 32 (Am. Math. Soc, Providence 1977).Google Scholar
7.Vaughan, R. C.. “Mean value theorems in prime number theory”, J. London Math. Soc. (2), 10 (1975), 153–162.CrossRefGoogle Scholar
8.Vaughan, R. C.. “On the distribution of αp modulo one”, Mathematika, 24 (1977), 136–141.CrossRefGoogle Scholar
9.Vaughan, R. C.. “An elementary method in prime number theory”, Acta Arithmetica, 37 (1980), 111–115.CrossRefGoogle Scholar
10.Vinogradov, I. M., “A new estimation of a trigonometric sum containing primes”, Bull. Acad. Sc. URSS Ser. Math., 2 (1938), 1–13.Google Scholar
11.Vinogradov, I. M.. “On the estimation of simplest trigonometric sums involving primes”, Izv. Akad. Nauk SSSR Ser. Mat., 2 (1939), 371–395.Google Scholar
12.Vinogradov, I. M.. “On the estimation of a trigonometric sum over primes”, Izv. Akad. Nauk SSSR Ser. Mat., 12 (1948), 225–248.Google Scholar