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 ($\rho \ne {\rho }_0$). 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.

This is the final published version, Cogent Mathematics, 2016

References in corpus (6)

• Integral and Series Representations of Riemann's Zeta function, Dirichelet's Eta Function and a Medley of Related Results
• A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis II
• Statistical and other properties of Riemann zeros based on an explicit equation for the $n$-th zero on the critical line
• A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis III. Exponential weight functions
• Some sums over the non-trivial zeros of the Riemann zeta function
• Notes on the Zeros of Riemann's Zeta Function

Cited by in corpus (3)

• A Note on the Riemann $\xi$-Function
• An Integral Equation for Riemann's Zeta Function and its Approximate Solution
• Uncovering functional relationships at zeros with special reference to Riemann's Zeta Function