Sums of quadratic characters.

Abstract

This thesis explores the question of determining the size of the sum$$S(X,Y) = \sum\sb{D\in{\cal Q}\atop\vert D\vert\le X}\left\vert\sum\sb{n\le Y}\left({D\over n}\right)\right\vert\sp2,$$where (D/n) is the Kronecker symbol and defines a quadratic character. In 1973 Jutila proved that S(X,Y) $\ll$ $XY\log\sp8X$. In 1975 Saparnijazov and Fainleib refined Jutila's method to obtain an upper bound of $XY\log\sp2X$; this was thought to be best possible. This thesis proves that in fact S(X,Y) $\ll$ $XY\log X$.