Solution to the Riemann Hypothesis from geometric analysis of component series functions in the functional equation of zeta

TL;DR

This paper claims to prove the Riemann Hypothesis by analyzing the geometric properties of series functions derived from the zeta function's functional equation, showing they cannot negate each other off the critical line.

Contribution

It introduces a novel geometric analysis of component series functions in the zeta function's functional equation to prove the hypothesis.

Findings

Series functions vanish at non-trivial zeros

Component functions cannot be additive inverses off the critical line

Proof of the Riemann Hypothesis based on geometric contradiction

Abstract

This paper presents a new approach towards the Riemann Hypothesis. On iterative expansion of integration term in functional equation of the Riemann zeta function we get sum of two series functions. At the `non-trivial' zeros of zeta function, value of the series is zero. Thus, Riemann hypothesis is false if that happens for an `s' off the line $\Re(s)=1/2$ ( the critical line). This series has two components $f(s)$ and $f(1-s)$. For the hypothesis to be false one component is additive inverse of the other. From geometric analysis of spiral geometry representing the component series functions $f(s)$ and $f(1-s)$ on complex plane we find by contradiction that they cannot be each other's additive inverse for any $s$, off the critical line. Thus, proving truth of the hypothesis.