Generalized lambert series, raabe's cosine transform and a generalization of ramanujan's formula for ζ (2m + 1)

A comprehensive study of the generalized Lambert series 0, and, is undertaken. Several new transformations of this series are derived using a deep result on Raabe's cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan's famous formula for for 0, the transformation formula for the logarithm of the Dedekind eta function and Wigert's formula for even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series and. An identity relating is obtained for odd and. In particular, this gives a beautiful relation between and. New results involving infinite series of hyperbolic functions with in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.