Experimental Mathematics

The Asymptotic Distribution of Exponential Sums, I

S. J. Patterson

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 {$S(f(x),c)$}.

Article information

Source
Experiment. Math., Volume 12, Issue 2 (2003), 135-153.

Dates
First available in Project Euclid: 31 October 2003