Further Exploration of Riemann's Functional Equation

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 $ζ(s)$ and $ζ^{\prime}(s)$ 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 $ζ(s)$ on opposite sides of the critical line, reducing to the analytic expression for $arg(ζ(1/2+iρ))$ first obtained in the previous work. Relationships are obtained between various combinations of $|ζ(s)|$ and $|ζ^{\prime}(s)|$, particularly on the critical line, and it is demonstrated that $arg(ζ(1/2+iρ))$ and $arg(ζ^{\prime}(1/2+iρ))$ uniquely define $|ζ(1/2+iρ)|$. A comment is made about the utility of such results as they might apply to putative proofs of Riemann's Hypothesis (RH).