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Convolution and Equidistribution by Nicholas M. Katz

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Introduction

The systematic study of character sums over finite fields may be said to have begun over 200 years ago, with Gauss. The Gauss sums over Fp are the sums

image

for à a nontrivial additive character of Fp, e.g., x image eix/p, and χ a nontrivial multiplicative character of F×p. Each has absolute value √p. In 1926, Kloosterman [Kloos] introduced the sums (one for each aF×p)

image

which bear his name, in applying the circle method to the problem of four ...

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