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

TL;DR

This paper develops transformation formulas for generalized Eisenstein series involving periodic functions, leading to new Ramanujan-type formulas for periodic zeta-functions and establishing reciprocity laws for related sums.

Contribution

It introduces new transformation formulas for Eisenstein series with periodic coefficients and derives Ramanujan-like formulas for periodic zeta-functions, extending classical results.

Findings

Derived transformation formulas for Eisenstein series with periodic functions.

Established reciprocity laws for periodic Apostol-Dedekind sums.

Obtained Ramanujan-type formulas for periodic zeta-functions.

Abstract

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.