Marie Jameson

Affiliation: Emory University

Email: mjames7@emory.edu

Title Of Talk: A Problem of Zagier on Quadratic Polynomials and Continued Fractions

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Abstract: For any non-square $1 < D\equiv 0,1 \text{ (mod }$4), Zagier defined \[ A_D(x) := \sum_{\begin{subarray}{c} \text{disc}(Q)=D\\ Q(\infty) < 0 < Q(x) \end{subarray}} Q(x) \] and proved that $A_D(x)$ is a constant function. For rational $x$, it turns out that this sum is finite. Here we address the infinitude of the number of quadratic polynomials for nonrational $x$, and more importantly address some problems posed by Zagier related to characterizing the polynomials which arise in terms of the continued fraction expansion of $x$. In addition, we study the divisibility of the constant functions $A_D(x)$ as $D$ varies, by using the Cohen-Eisenstein series and various Hecke-type operators.

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