Generalized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for $\zeta (2m+1)$

A comprehensive study of the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}\exp{(-an^{N}x)}}{1-\exp{(-n^{N}x)}}, 0<a\leq 1,\ x>0$,$N\in$\mathbb{N}$$and$h\in$\mathbb{Z}$$, is undertaken. Two of the general transformations of this series that we obtain here lead to two-parameter generalizations of Ramanujan’s famous formula for $\zeta (2m+1)$, $m>0$ and the transformation formula for $\log \eta (z)$. Numerous important special cases of our transformations are derived. An identity relating $\zeta (2N+1),\zeta (4N+1),\cdots ,\zeta (2Nm+1)$ is obtained for $N$ odd and $m\in \mathbb{N}$. Certain transcendence results of Zudilin- and Rivoal-type are obtained for odd zeta values and generalized Lambert series. A criterion for transcendence of $\zeta (2m+1)$ and a Zudilin-type result on irrationality of Euler’s constant $\gamma$ are also given. New results analogous to those of Ramanujan and Klusch for $N$ even, and a transcendence result involving $\zeta (2m+1-\frac{1}{N})$, are obtained.