Deformations of $\Xi(s)=\Xi(1-s)$ and the heat equation

Abstract

This paper studies deformations of the well-known $\Xi(s)=\Xi(1-s)$ equation for the Riemann $\Xi$ function satisfied by $\Xi_\rho(s):=\int_{0}^\infty\frac{{\d}t}{t}{t^{\frac{s}{2}}}\Psi(t)e^{-\rho\ln^2(t)}$ where ${\Psi(t)=\sum_{n\in\mathbb{N}^+}e^{-\pi{n}^2t}}$.