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

A comprehensive study of the generalized Lambert series ?n=1?nN?2hexp(?anNx)1?exp(?nNx),0<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 ?(2m+1), m>0 and the transformation formula for log?(z). Numerous important special cases of our transformations are derived. An identity relating ?(2N+1),?(4N+1),?,?(2Nm+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 ?(2m+1) and a Zudilin-type result on irrationality of Euler's constant ? are also given. New results analogous to those of Ramanujan and Klusch for N even, and a transcendence result involving ?(2m+1?1N), are obtained.