Exploring Riemann's Functional Equation

Michael Milgram

Published 2015-10-20Version 1

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 equivalent to RH as claimed, but RH is not proven. It is proven that the order of a zero on the critical line is never even, and that the derivative $\zeta^{\prime}(1/2+i\rho)$ will never vanish on the punctured critical line ($\rho\neq\rho_0$), nor at a simple zero. Traditional asymptotic and counting results are obtained in an untraditional manner, yielding insight into the nature of $\zeta(s)$ on the critical line.