Open Access
2003 The Asymptotic Distribution of Exponential Sums, I
S. J. Patterson
Experiment. Math. 12(2): 135-153 (2003).


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


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S. J. Patterson. "The Asymptotic Distribution of Exponential Sums, I." Experiment. Math. 12 (2) 135 - 153, 2003.


Published: 2003
First available in Project Euclid: 31 October 2003

zbMATH: 1061.11046
MathSciNet: MR2016703

Primary: 11L05
Secondary: 11N37 , 11Y35

Keywords: Arithmetic functions , Complete exponential sums , Gauss sums , Linnik-Selberg conjecture

Rights: Copyright © 2003 A K Peters, Ltd.

Vol.12 • No. 2 • 2003
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