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

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A comprehensive study of the generalized Lambert series ∑ₙ₌₁∞nⁿ⁻²ʰ(-anⁿx)1-(-nⁿx), 0<a≤ 1,x>0, N and h, 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 ζ(2m+1), m>0 and the transformation formula for η(z). Numerous important special cases of our transformations are derived. An identity relating ζ(2N+1), ζ(4N+1),, ζ(2Nm+1) is obtained for N odd and m. Certain transcendence results of Zudilin- and Rivoal-type are obtained for odd zeta values and generalized Lambert series. A criterion for transcendence of ζ(2m+1) and a Zudilin-type result on irrationality of Euler's constant γ are also given. New results analogous to those of Ramanujan and Klusch for N even, and a transcendence result involving ζ(2m+1-1/N), are obtained.