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The Estimation of Complete Exponential Sums

Published online by Cambridge University Press:  20 November 2018

J. H. Loxton and
R. C. Vaughan
J. H. Loxton
Affiliation:
University of New South Wales Sydney, Australia
R. C. Vaughan
Affiliation:
Imperial College London, England
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Abstract

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This paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πi f(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn (q)qe/(e+1) , where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.

Keywords

10G1012B05

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 4 , 01 December 1985, pp. 440 - 454
Copyright
Copyright © Canadian Mathematical Society 1985

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