Exploring Riemann's Functional Equation

Abstract

An equivalent, but variant form of the Riemann functional equation is explored, and several discoveries are made. Properties of the Riemann zeta function $\zeta(s)$ from which a necessary and sufficient condition for the existence of zeros in the critical strip are deduced. This in turn, by an indirect route, eventually produces a simple, solvable, differential equation for $arg(\zeta(s))$ on the critical line $s=1/2+i\rho$, the consequences of which are explored, and the "LogZeta" function is introduced. A singular linear transform between the real and imaginary components of $\zeta$ and $\zeta^\prime$ on the critical line is derived, and an implicit relationship for locating a zero ($\rho=\rho_0$) on the critical line is found between the arguments of $\zeta(1/2+i\rho)$ and $\zeta^{\prime}(1/2+i\rho)$. Notably, the Volchkov criterion, a Riemann Hypothesis (RH) equivalent is analytically evaluated and verified to be half equivalent to RH, but RH is not proven. Numerical results are presented, some of which lead to the identification of {\it anomalous zeros}, whose existence in turn suggests that well-established, traditional derivations such as the Volchkov criterion and counting theorems require re-examination. It is proven that the derivative $\zeta^{\prime}(1/2+i\rho)$ will never vanish on the perforated critical line ($\rho\neq\rho_0$). Traditional asymptotic and counting results are obtained in an untraditional manner, yielding insight into the nature of $\zeta(1/2+i\rho)$ as well as very accurate asymptotic estimates for distribution bounds and the density of zeros on the critical line.