A problem of Zagier on quadratic polynomials and continued fractions

Abstract

For non-square [Formula: see text] (mod 4), Don Zagier defined a function [Formula: see text] by summing over certain integral quadratic polynomials. He proved that [Formula: see text] is a constant function depending on [Formula: see text]. For rational [Formula: see text], it turns out that this sum has finitely many terms. Here we address the infinitude of the number of quadratic polynomials for non-rational [Formula: see text], and more importantly address some problems posed by Zagier related to characterizing the polynomials which arise in terms of the continued fraction expansion of [Formula: see text]. In addition, we study the indivisibility of the constant functions [Formula: see text] as [Formula: see text] varies.