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

TL;DR

This paper explores integral and series transformations linked to Ramanujan's identities, introduces new transformations, and establishes Salem-type equivalences to the Riemann hypothesis, advancing understanding of these complex mathematical relationships.

Contribution

It introduces new integral and series transformations based on Ramanujan's identities and establishes Salem-type equivalences to the Riemann hypothesis.

Findings

New transformations like Widder-Lambert and Kontorovich-Lebedev are demonstrated.

Reciprocal inversion formulas are proved in a specific Banach space.

Salem-type equivalences to the Riemann hypothesis are established.

Abstract

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.