标题： 关于广义的艾森斯坦级数和拉马努金的周期zeta函数公式 显示英文标题

标题： On generalized Eisenstein series and Ramanujan's formula for periodic zeta-functions

摘要： 在本文中，针对由\[ G(z,s;A_{\alpha},B_{\beta};r_{1},r_{2})=\sum\limits_{m,n=-\infty}^{\infty }\ \hspace{-0.19in}^{^{\prime}}\frac{f(\alpha m)f^{\ast}(\beta n)} {((m+r_{1})z+n+r_{2})^{s}},\text{ }\operatorname{Re}(s)>2,\text{ }\operatorname{Im}(z)>0 \]定义的大量Eisenstein级数在$s=1-r$，$r\in\mathbb{N}$的变换公式被研究。 这里 $\left\{ f(n)\right\}$ 和 $\left\{ f^{\ast}(n)\right\}$, $-\infty<n<\infty$是具有周期 $k>0$的复数序列，而 $A_{\alpha}=\left\{ f(\alpha n)\right\} $ 和 $B_{\beta}=\left\{ f^{\ast}(\beta n)\right\}$, $\alpha,\beta\in\mathbb{Z}$。 在变换公式中出现的是涉及周期伯努利函数的Dedekind和的推广形式。 在变换公式的上下文之外，证明了周期Apostol-Dedekind和的互易定律。 此外，给出了$G(z,s;A_{\alpha},I;r_{1},r_{2})$和$G(z,s;I,A_{\alpha };r_{1},r_{2})$的变换公式，其中$I=\left\{ 1\right\}$。作为这些公式的应用，导出了周期zeta函数的拉马努金公式的类似形式。 显示英文摘要

摘要： In this paper, transformation formulas for a large class of Eisenstein series defined by \[ G(z,s;A_{\alpha},B_{\beta};r_{1},r_{2})=\sum\limits_{m,n=-\infty}^{\infty }\ \hspace{-0.19in}^{^{\prime}}\frac{f(\alpha m)f^{\ast}(\beta n)} {((m+r_{1})z+n+r_{2})^{s}},\text{ }\operatorname{Re}(s)>2,\text{ }\operatorname{Im}(z)>0 \] are investigated for $s=1-r$, $r\in\mathbb{N}$. Here $\left\{ f(n)\right\}$ and $\left\{ f^{\ast}(n)\right\}$, $-\infty<n<\infty$ are sequences of complex numbers with period $k>0$, and $A_{\alpha}=\left\{ f(\alpha n)\right\} $ and $B_{\beta}=\left\{ f^{\ast}(\beta n)\right\}$, $\alpha,\beta\in\mathbb{Z}$. Appearing in the transformation formulas are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity law is proved for periodic Apostol-Dedekind sum outside of the context of the transformation formulas. Furthermore, transformation formulas are presented for $G(z,s;A_{\alpha},I;r_{1},r_{2})$ and $G(z,s;I,A_{\alpha };r_{1},r_{2})$, where $I=\left\{ 1\right\}$. As an application of these formulas, analogues of Ramanujan's formula for periodic zeta-functions are derived.