  We consider integral and series transformations, which are associated with
Ramanujan's identities, involving various arithmetic functions and a ratio of
products of Riemann's zeta functions of different arguments. Reciprocal
inversion formulas are proved in a Banach space of functions whose Mellin's
transforms are integrable over the vertical line Re s > 1. Examples of new
transformations like Widder-Lambert and Kontorovich-Lebedev type are exhibited.
Particular cases include familiar Lambert and Mobius transformations. Finally
a class of equivalences of the Salem type to the Riemann hypothesis is
established.
