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

Abstract

We consider integral and series transformations, which are associated with Ramanujan's identities, involving arithmetic functions a(n), ω(n), σa(n), d(n), μ(n), λ(n), ϕ(n) 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 Möbius transformations. Finally, a class of equivalences of the Salem type to the Riemann hypothesis is established.