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

TL;DR

This paper explores deformations of the Riemann Xi function's functional equation using a heat equation-inspired approach, introducing a parameterized family of functions and analyzing their properties.

Contribution

It introduces a new family of deformed Xi functions via a heat equation framework, extending the classical functional equation of the Riemann Xi function.

Findings

Defined the deformed Xi functions with a parameter rho.

Analyzed the properties and symmetries of these deformed functions.

Connected the deformations to solutions of a 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 ${\Psi(t)=\sum_{n\in\mathbb{N}^+}e^{-\pi{n}^2t}}$.