On curious properties of the general size-biased distribution

TL;DR

This paper explores unique properties of a size-biased distribution connected to the Riemann xi-function, revealing intriguing functional equations and implications for the Riemann hypothesis.

Contribution

It provides new insights into the size-biased distribution's properties and their relation to the Riemann xi-function and the Riemann hypothesis.

Findings

Identifies curious properties of the size-biased distribution

Links these properties to special functional equations

Relates observations to recent Riemann hypothesis developments

Abstract

We offer further results on a general size-biased distribution related to the Riemann xi-function we presented in [9] using the work of Ferrar. Curious properties associated with its expected value are presented, which are related to special functional equations. We also relate our observations to some recent developments related to the Riemann hypothesis.