A comprehensive study of the generalized Lambert series0 < a ≤ 1, x > 0, N ∈ N and h ∈ 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 ζ(2m + 1), m > 0 and the transformation formula for log η(z). Numerous important special cases of our transformations are derived. An identity relating ζ(2N + 1), ζ(4N + 1), · · · , ζ(2N m + 1) is obtained for N odd and m ∈ N. 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.

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