An explicit formula for the fourth moment of certain exponential sums
type: Item
instance of: scholarly article
title: An explicit formula for the fourth moment of certain exponential sums
zbMATH Open document ID: 1240.11090
author: Catalina Calderón
author: M. J. de Velasco
author: M. J. Zárate
published in: Acta Mathematica Hungarica
publication date: 2011-07-22T00:00:00Z
review text: Consider the generalised Kloosterman sum \[ S(m,n,\chi,\chi',q) = \mathop{{\sum}'}_{a=1}^q \chi(a)G(a,\chi') e\left({ma^k+na\over q}\right) \] for integers \(m, n, q, k\) with \(q, k\geq 1\) and Dirichlet characters \(\chi,\chi'\bmod q\) and the Gaussian sum \[ G(a,\chi')=\mathop{{\sum}'}_{u=1}^q \chi'(u)e\left({ua\over q}\right). \] The main result is an exact formula for the fourth moment \(M_k(q)\) of \(S(m,n,\chi,\chi',q)\) when averaged over the parameters \(m, \chi, \chi'\). The authors note that the sums are closely related to certain mixed exponential sums considered by \textit{H. N. Liu} [Proc. Am. Math. Soc. 136, 1193--1203 (2008; Zbl 1145.11063)] because \(S(m,n,\chi,\chi',p^\alpha)= G(1,\chi')C(m,n,\chi\overline{\chi' }, k,p^\alpha)\) where \(C\) is Liu's sum \[ C(m,n,\chi,k,q)=\mathop{{\sum}'}_{a=1}^q \chi(a) e\left({ma^k+na\over q}\right) \] and the theorem also follows from Liu's result giving the fourth moment of \(C(m, n, \chi, k,q)\) averaged over \(\chi\) and \(m\).
reviewed by: John H. Loxton
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 11L03
zbMATH DE Number: 5931411
zbMATH Keywords: exponential sum
zbMATH Keywords: Gaussian sums, Kloosterman sums, fourth power mean
MaRDI profile type: Publication
full work available at URL: s10474-010-0043-5
OpenAlex ID: W1970565332
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DOI: 10.1007/S10474-010-0043-5
description: scientific article; zbMATH DE number 5931411