## Experimental Mathematics

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

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

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 31 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1067634728

**Mathematical Reviews number (MathSciNet)**

MR2016703

**Zentralblatt MATH identifier**

1061.11046

**Subjects**

Primary: 11L05: Gauss and Kloosterman sums; generalizations

Secondary: 11N37: Asymptotic results on arithmetic functions 11Y35: Analytic computations

**Keywords**

Complete exponential sums Gauss sums arithmetic functions Linnik-Selberg conjecture

#### Citation

Patterson, S. J. The Asymptotic Distribution of Exponential Sums, I. Experiment. Math. 12 (2003), no. 2, 135--153. https://projecteuclid.org/euclid.em/1067634728