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Some remarks on Gauss sums associated with kth powers

Published online by Cambridge University Press:  24 October 2008

H. L. Montgomery ,
R. C. Vaughan  and
T. D. Wooley
H. L. Montgomery
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109–1003, U.S.A.
R. C. Vaughan
Affiliation:
Department of Mathematics, Imperial College of Science and Technology, Queen's Gate, London SW7 2BZ
T. D. Wooley
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109–1003, U.S.A.
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Extract

Estimates for rational trigonometric sums are of great importance in analysing the local aspects of many additive problems. Indeed, bounds for the sums

in which e(α) denotes exp (2πiα), play a fundamental rôle in the application of the Hardy–Littlewood method to Waring's problem (see [11]), and also in the analysis of the local solubility of systems of additive equations (see, for example, [2]). When k ≥ 2 is an integer, and p is a prime number it is well known (see [5] or [11, lemma 4·3]) that

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 1 , July 1995 , pp. 21 - 33
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

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