# Sums of quadratic functions with two discriminants

## TL;DR

This paper generalizes Zagier's construction of functions related to modular forms to include both even and odd weights, introducing a two-discriminant function whose average value links to Fourier coefficients of Eisenstein series.

## Contribution

It extends Zagier's quadratic function construction to odd weights and incorporates two discriminants, broadening the scope of modular form connections.

## Key findings

- The new function $F_{k,D,d}(x)$ generalizes Zagier's function for both even and odd $k$.
- The average value of the new function corresponds to Fourier coefficients of Eisenstein series.
- The construction reduces to Zagier's original function when $d=1$.

## Abstract

Zagier in [4] discusses a construction of a function $F_{k,D}(x)$ defined for an even integer $k \geq 2$, and a positive discriminant $D$. This construction is intimately related to half-integral weight modular forms. In particular, the average value of this function is a constant multiple of the $D$-th Fourier coefficient of weight $k+1/2$ Eisenstein series constructed by H. Cohen in \cite{Cohen}. In this note we consider a construction which works both for even and odd positive integers $k$. Our function $F_{k,D,d}(x)$ depends on two discriminants $d$ and $D$ with signs sign$(d)=$ sign$(D)=(-1)^k$, degenerates to Zagier's function when $d=1$, namely, \[ F_{k,D,1}(x)=F_{k,D}(x), \] and has very similar properties. In particular, we prove that the average value of $F_{k,D,d}(x)$ is again a Fourier coefficient of H. Cohen's Eisenstein series of weight $k+1/2$, while now the integer $k \geq 2$ is allowed to be both even and odd.