  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.
