Exploring Riemann's Functional Equation

TL;DR

This paper explores a variant form of Riemann's functional equation, deriving new properties and relationships of the zeta function, and provides numerical evidence and insights into the distribution of zeros, including potential anomalies.

Contribution

It introduces a new differential equation for the argument of the zeta function, derives a linear transform between components of zeta and its derivative, and evaluates the Volchkov criterion in relation to the Riemann Hypothesis.

Findings

A necessary and sufficient condition for zeros in the critical strip is deduced.

A simple, solvable differential equation for arg(zeta) on the critical line is produced.

Numerical results suggest the existence of anomalous zeros, challenging traditional assumptions.

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.