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

Abstract

In this paper, transformation formulas for a large class of Eisenstein series defined by \[ G(z,s;A_α,B_β;r_{1},r_{2})=\sum\limits_{m,n=-\infty}^{\infty }\ \hspace{-0.19in}^{^{\prime}}\frac{f(αm)f^{\ast}(βn)} {((m+r_{1})z+n+r_{2})^{s}},\text{ }\operatorname{Re}(s)>2,\text{ }\operatorname{Im}(z)>0 \] 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.

arXiv admin note: text overlap with arXiv:1506.01809