Experimental Mathematics

The Asymptotic Distribution of Exponential Sums, I

S. J. Patterson

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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


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