A Note on the Riemann $\xi$-Function

TL;DR

This paper explores integrals and integral equations related to Riemann's 1-function, offering a new derivation of a key identity and discussing the significance of the line s=3/2+it in the critical strip.

Contribution

It provides a less restrictive derivation of a key identity for the Riemann 1-function and introduces the idea of a holographic image of the critical strip on a specific line.

Findings

New derivation of a key identity for 1-function

Identification of the line s=3/2+it as containing a holographic image of the critical strip

Insights into the integral equations satisfied by 1-function

Abstract

This note investigates a number of integrals of and integral equations satisfied by Riemann's $\xi-$function. A different, less restrictive, derivation of one of his key identities is provided. This work centers on the critical strip and it is argued that the line $s=3/2+i t$ , e.g., contains a holographic image of this region.