Eisenstein-series evaluations for a family of hyperbolic cosine Lambert series

TL;DR

This paper evaluates a family of hyperbolic Lambert series and related hyperbolic series using Eisenstein series derivatives at CM points, providing explicit identities and a modular framework for these hyperbolic cosine identities.

Contribution

It introduces a novel method linking hyperbolic Lambert series to derivatives of Eisenstein series at CM points, leading to explicit evaluations and a modular explanation for hyperbolic cosine identities.

Findings

Explicit evaluations of series S_0, S_1, and vanishing of S_m for m>1.

Evaluation of a quadratic hyperbolic series with a closed-form expression.

Connection of hyperbolic series to Eisenstein series derivatives and modular forms.

Abstract

We study a family of hyperbolic Lambert series of the form \[ S_m=\sum_{n=1}^\infty\left( \frac{n^{2m}}{\cosh(\pi n)-1} -\frac{(2^{2m+1}-(-1)^{m(m+1)/2}2^{m+1}+4) n^{2m}}{\cosh(2\pi n)-1} +\frac{2^{2m+2}n^{2m}}{\cosh(4\pi n)-1} \right). \] We prove that \[ S_0=\frac1{12},\qquad S_1=\frac1{2\pi^2},\qquad S_m=0 \quad (m>1). \] We also evaluate the quadratic hyperbolic series \[ \sum_{n=1}^\infty \left( \frac{4}{(\cosh(\pi n)-1)^2} -\frac{55}{(\cosh(2\pi n)-1)^2} +\frac{16}{(\cosh(4\pi n)-1)^2} \right) = \frac{77-234/\pi}{72}. \] The proof is based on rewriting the hyperbolic kernels as Lambert series and identifying the resulting sums with derivatives of Eisenstein series at the CM points $i/2$, $i$, and $2i$. The initial evaluations are reduced to explicit identities for $E_2$, $E_4$, and $E_6$, together with a theta-constant relation at $2i$, while the general vanishing result is obtained from a parity decomposition of the Gaussian lattice in the classical Eisenstein series $G_k$. This gives a uniform modular explanation for a family of hyperbolic cosine identities.