A Note on the $S_2(\delta)$ Distribution and the Riemann Xi Function

Abstract

The theory of $S_2(\delta)$ family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The $\delta$ deformation of the xi function is formulated in terms of the $S_2(\delta)$ distribution and shown to satisfy Riemann's functional equation. Criteria for simplicity of roots of the xi function and for its simple roots to satisfy the Riemann hypothesis are formulated in terms of a differentiability property of the $S_2(\delta)$ family. For application, the values of the Riemann zeta function at the integers and of the Riemann xi function in the complex plane are represented as integrals involving the Laplace transform of $S_2.$