A New Integral Equation and Some Integrals Associated with Number Theory

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.