A New Integral Equation and Some Integrals Associated with Number Theory

TL;DR

This paper introduces a novel integral equation linked to the Riemann xi-function, offering new proofs of classical identities, recasting RH criteria, and applying the equation to the Dirichlet problem in the half-plane.

Contribution

It presents a new integral equation involving the Riemann xi-function, combining integral transforms with classical number theory results, and applies it to solve a Dirichlet problem.

Findings

Derived a new integral equation related to the Riemann xi-function

Provided a new proof of existing functional identities in number theory

Applied the integral equation to the Dirichlet problem in the half-plane

Abstract

We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann $\xi$-function, to obtain a new integral equation. We also provide a new proof of known functional-type identities from analytic number theory, and recast some criteria associated with the RH. We also describe an application of our integral equation to the Dirichlet problem in the half plane, giving a new application of the Riemann xi function.