A Reformulation of the Xi Function

TL;DR

This paper introduces a new formulation of the Riemann Xi function using incomplete gamma functions, providing insights that support the Riemann hypothesis by analyzing zero level curves.

Contribution

It presents a novel reformulation of the Xi function and uses it to analyze zeros, offering a new approach to validate the Riemann hypothesis.

Findings

Zeros off the critical line are inconsistent with the properties of the reformulated Xi function.

The reformulation supports the Riemann hypothesis by showing zeros lie on the critical line.

Properties of the reformulated function help analyze the intersection of zero level curves.

Abstract

This paper proposes a reformulation of the Riemann Xi function in order to investigate its properties. The reformulated function, which depicts the Xi function as the weighted sum of incomplete gamma functions, is validated, and a number of properties are established. These properties are then used to analyze the intersection of the real and imaginary zero level curves. It is shown that a pair of zeros off the critical line is not consistent with these properties, thus validating the Riemann hypothesis.