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

Abstract

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 ζ(2m+1), m>0 and the transformation formula for logη(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∈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+1−1N), are obtained.