On character sums of binary quadratic forms
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 \(p\).
scientific article; zbMATH DE number 5575277