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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 ofthe general transformations of this series that we obtain here lead totwo-parameter generalizations of Ramanujan's famous formula for $\zeta(2m+1)$,$m>0$ and the transformation formula for $\log\eta(z)$. Numerous importantspecial 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 areobtained for odd zeta values and generalized Lambert series. A criterion fortranscendence of $\zeta(2m+1)$ and a Zudilin-type result on irrationality ofEuler's constant $\gamma$ are also given. New results analogous to those ofRamanujan and Klusch for $N$ even, and a transcendence result involving$\zeta\left(2m+1-\frac{1}{N}\right)$, are obtained.

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