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We consider integral and series transformations, which are associated withRamanujan's identities, involving various arithmetic functions and a ratio ofproducts of Riemann's zeta functions of different arguments. Reciprocalinversion formulas are proved in a Banach space of functions whose Mellin'stransforms are integrable over the vertical line Re s > 1. Examples of newtransformations like Widder-Lambert and Kontorovich-Lebedev type are exhibited.Particular cases include familiar Lambert and Mobius transformations. Finallya class of equivalences of the Salem type to the Riemann hypothesis isestablished.

Integral and series transformations via Ramanujan's identities and Salem's type equivalences to the Riemann hypothesis