Further Exploration of Riemann's Functional Equation

TL;DR

This paper extends the analysis of Riemann's functional equation from the critical line to the entire complex plane, providing simpler derivations, new relationships, and potential implications for the Riemann Hypothesis.

Contribution

It introduces a simplified derivation of the fundamental equation and generalizes key relationships between zeta function components across the complex plane.

Findings

Derived a simpler relationship between real and imaginary parts of (s) and '(s)

Obtained an analytic expression for the sum of arguments of (s) on opposite sides of the critical line

Showed that argument functions and magnitudes of (s) and '(s) uniquely determine each other on the critical line

Abstract

A previous exploration of the Riemann functional equation that focussed on the critical line, is extended over the complex plane. Significant results include a simpler derivation of the fundamental equation developed previously, and its generalization from the critical line to the complex plane. A simpler statement of the relationship that exists between the real and imaginary components of $\zeta(s)$ and $\zeta^{\prime}(s)$ on opposing sides of the critical line is developed, reducing to a simpler statement of the same result on the critical line. An analytic expression is obtained for the sum of the arguments of $\zeta(s)$ on opposite sides of the critical line, reducing to the analytic expression for $arg(\zeta(1/2+i\rho))$ first obtained in the previous work. Relationships are obtained between various combinations of $|\zeta(s)|$ and $|\zeta^{\prime}(s)|$, particularly on the critical line, and it is demonstrated that $arg(\zeta(1/2+i\rho))$ and $arg(\zeta^{\prime}(1/2+i\rho))$ uniquely define $|\zeta(1/2+i\rho)|$. A comment is made about the utility of such results as they might apply to putative proofs of Riemann's Hypothesis (RH).