The Asymptotic Distribution of Exponential Sums, I

Abstract

Let {$f(x)$} be a polynomial with integral coefficients and let, for {$c>0$, $S(f(x),c)=\sum_{j \pmod c} \exp(2\pi\imath\frac{f(j)}c)$}. It has been possible, for a long time, to estimate these sums efficiently. On the other hand, when the degree of {$f(x)$} is greater than 2 very little is known about their asymptotic distribution, even though their history goes back to C. F. Gauss and E. E. Kummer. The purpose of this paper is to present both experimental and theoretic evidence for a very regular asymptotic behaviour of {$S(f(x),c)$}.