An Arithmetic Resolvent and Heat Kernel for the Riemann Xi Function in the Variable \(s(1-s)\)

Abstract

We descend the completed Riemann Xi function through the quotient map $s\mapsto s(1-s)$ and obtain an entire function $\mathcal D$ of order $1/2$ and genus zero. Its zeros are the functional-equation orbit values $\rho(1-\rho)$, and the Riemann hypothesis is equivalent to their support on $[1/4,\infty)$. The logarithmic derivative on the negative axis yields an arithmetic resolvent that admits both an explicit prime--Gamma expression and a zero-orbit Stieltjes expansion. We then construct an arithmetic heat kernel whose prime term is a Gaussian in $\log n$ and prove an exact Laplace transform identity linking the heat kernel to the resolvent. Finally, we identify the same heat kernel with the normally convergent zero-orbit heat sum. All interchanges of series and integrals are justified without assuming the Riemann hypothesis. The paper packages classical ingredients into a single quotient-variable resolvent/heat formalism.