Uncovering functional relationships at zeros with special reference to Riemann's Zeta Function

TL;DR

This paper explores a Master equation's properties, extending its applicability to analyze the Riemann Zeta function's zeros and derive sum rules, demonstrating its utility for understanding functional relationships.

Contribution

It introduces an extended form of a Master equation applicable to a broader parameter range and demonstrates its use with the Riemann Zeta function to derive sum rules among zeros.

Findings

Extended the Master equation's applicability to a wider parameter range.

Derived sum rules among the zeros of the Riemann Zeta function.

Demonstrated the utility of the equation with various examples.

Abstract

A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending its applicability to a general range of an independent parameter. Examples are given for various values of the parameter using Riemann's Zeta function as a template to demonstrate the utility of the equation. The template is then extended to the derivation of various sum rules among the zeros of the Zeta function as an example of how similar rules can be obtained for other functions.