Zagier in [4] discusses a construction of a function $F_{k,D}(x)$ defined foran even integer $k \geq 2$, and a positive discriminant $D$. This constructionis intimately related to half-integral weight modular forms. In particular, theaverage value of this function is a constant multiple of the $D$-th Fouriercoefficient 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 oddpositive 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'sfunction when $d=1$, namely, \[ F_{k,D,1}(x)=F_{k,D}(x), \] and has verysimilar 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 seriesof weight $k+1/2$, while now the integer $k \geq 2$ is allowed to be both evenand odd.
展开▼
机译：Zagier在[4]中讨论了函数$ F_ {k，D}（x）$的构造，该函数为偶数整数$ k \ geq 2 $和正判别式$ D $定义。这种构造与半整体重量模块形式密切相关。特别是，此函数的平均值是H. Cohen in \ cite {Cohen}构造的第$ D $次傅立叶权重$ k + 1/2 $ Eisenstein级数的常数倍。在本说明中，我们考虑一种对偶数和奇数整数$ k $都有效的构造。我们的函数$ F_ {k，D，d}（x）$取决于两个判别式$ d $和$ D $，其符号为sign $（d）= $ sign $（D）=（-1）^ k $，简并当$ d = 1 $，即\ [F_ {k，D，1}（x）= F_ {k，D}（x），\]时具有Zagier函数，并且具有非常相似的性质。特别是，我们证明了$ F_ {k，D，d}（x）$的平均值再次是H. Cohen的权重$ k + 1/2 $的Eisenstein级数的傅立叶系数，而现在是整数$ k \ geq 2 $可以是偶数和奇数。
展开▼
