Zeta Functional Analysis

Abstract

We intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also establishes and presents new expository notes and perspectives on zeta function theory and functional analysis. In addition, a new fundamental result, in form of a new function called omega $\Omega(s)$, is introduced to analytic number theory for the first time. This new function together with some of its most fundamental properties and other related identities are here disclosed and presented as a new approach to the analysis of sums of generalised harmonic series, related alternating series and polygamma functions associated with Riemann zeta function.