On character sums of binary quadratic forms
type: Item
instance of: scholarly article
title: On character sums of binary quadratic forms
zbMATH Open document ID: 1192.11053
author: Mei-Chu Chang
published in: Journal of Number Theory
publication date: 2009-07-06T00:00:00Z
review text: Let \(f(x,y)=x^2+ky^2\) be an integral binary quadratic form, and \(\chi\) a nontrivial Dirichlet character modulo a prime number \(p\). The author studies the partial character sum \[ S(f)=\sum\limits_{(x,y)\in B}\chi(f(x,y)), \] where \[ B=\{(x,y)\in\mathbb{Z}^2\mid a\leq x\leq a+H, b\leq y\leq b+H\}, |a|,|b|<H<p. \] The author proves that, given \(\varepsilon>0\), there is \(\tau>0\) such that \[ |S(f)|<p^{-\tau}H^2,\quad H>p^{\frac14+\varepsilon} \] provided that \(p\) is sufficiently. The argument is a variant of Burgess' method. It goes as follows. Applying a technique possibly originated by H. Weyl, the author approximates the character sum by the mean value of its deformations \[ S(f(x+ut,y+vt), 1\leq t\leq M, (u,v)\in\mathfrak{D}\subset [0,D]^2. \] Changing the order of summation and applying Hölder's inequality and Weil's bound the author reduces the question to estimating the number of a subset of solutions of a system of congruences modulo \(p\).
reviewed by: Chunlei Liu
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 5575277
zbMATH Keywords: partial character sums
zbMATH Keywords: quadratic residues
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.02.017
OpenAlex ID: W1973419323
cites work: On Character Sums and Primitive Roots†
cites work: On the Quadratic Character of a Polynomial
cites work: A Note on Character Sums of Binary Quadratic Forms
cites work: Q4043134
cites work: Character Sums and Primitive Roots in Finite Fields
DOI: 10.1016/J.JNT.2009.02.017
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description: scientific article; zbMATH DE number 5575277