Koshliakov kernel and identities involving the Riemann zeta function

TL;DR

This paper explores integral identities involving the Riemann zeta function and Bessel-related kernels, deriving special cases linked to Ramanujan and Guinand transformations.

Contribution

It introduces new integral identities involving the Riemann zeta function and Bessel function kernels, connecting to classical transformations.

Findings

Derived new integral identities involving the Riemann zeta function and Bessel kernels.

Identified special cases related to Ramanujan and Guinand transformations.

Enhanced understanding of the interplay between zeta function and Bessel function identities.

Abstract

Some integral identities involving the Riemann zeta function and functions reciprocal in a kernel involving the Bessel functions $J_{z}(x), Y_{z}(x)$ and $K_{z}(x)$ are studied. Interesting special cases of these identities are derived, one of which is connected to a well-known transformation due to Ramanujan, and Guinand.