发表机构
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文仿照Ishibashi方法推导Zagier型zeta函数$\tilde{\mathcal{Z}}$在$s=1$处的全部Laurent系数，并研究其中涉及的函数$\mathfrak{F}_k^0(x)$的函数方程、Hecke算子作用及积分联系。
AI 中文摘要
Don Zagier的显著贡献之一是对实二次域的Kronecker极限公式，其中他将二重级数$\mathcal{Z}(s,w,w^\prime)$与实二次域相关的Dedekind zeta函数联系起来。后来，Ishibashi确定了$\mathcal{Z}(s,w,w^\prime)$在$s=1$处的所有Laurent系数。最近，Choie和kumar研究了类似二重级数$\tilde{\mathcal{Z}}(s,w,w^\prime)$的解析行为。在本文中，我们仿照Ishibashi的方法，推导出$\tilde{\mathcal{Z}}(s,w,w^\prime)$的所有Laurent系数。这些Laurent系数涉及一个有趣的函数$\mathfrak{F}_k^0(x)$，该函数此前由Dixit等人研究过（其中$k=1$时称为Ramanujan情形），他们获得了$\mathfrak{F}_k^0(x)$的一个优美的对称关系。我们建立了$\mathfrak{F}_k^0(x)$的两项和三项函数方程，推导了周期型Hecke算子对$\mathfrak{F}_k^0(x)$的作用，并将一个重要的积分与$\mathfrak{F}_k^0(x)$联系起来。
英文摘要
One of the remarkable contributions of Don Zagier was the Kronecker limit formula for a real quadratic field, where he connects the double series $\mathcal{Z}(s,w,w^\prime)$ to the Dedekind zeta function associated to a real quadratic field. Later, Ishibashi determined all the Laurent coefficients of $\mathcal{Z}(s,w,w^\prime)$ at $s=1$. Recently, Choie and kumar have studied the analytic behaviour of the analogous double series $\tilde{\mathcal{Z}}(s,w,w^\prime)$. In this article, we derive all the Laurent coefficients of $\tilde{\mathcal{Z}}(s,w,w^\prime)$, akin to Ishibashi. These Laurent coefficients involve an interesting function $\mathfrak{F}_k^0(x)$, which was earlier studied by Dixit et. al. (Ramanujan for $k=1$), where they obtained a beautiful symmetric relation for $\mathfrak{F}_k^0(x)$. We establish both the two term and the three term functional equation of $\mathfrak{F}_k^0(x)$, derive the action of the period-like Hecke operator on $\mathfrak{F}_k^0(x)$ and connect an important integral with $\mathfrak{F}_k^0(x)$.
Comments23 pages
