Zeta Functional Analysis

TL;DR

This paper explores deep connections between the Riemann zeta and gamma functions, introduces a new formula for odd zeta values, and presents a novel function Omega for advanced analysis in number theory.

Contribution

It introduces the new function Omega(s) and provides a novel approach to analyzing zeta function values and related series, expanding theoretical understanding.

Findings

Derived a new formula for (2n+1) in terms of fractional zeta values

Introduced the Omega(s) function with fundamental properties

Presented new perspectives on zeta function theory and 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.