Analysis of the Riemann Zeta Function via Recursive Taylor Expansions

TL;DR

This paper provides an unconditional proof that all non-trivial zeros of the Riemann Zeta function lie on the critical line by using recursive Taylor expansions and logical contradiction.

Contribution

It introduces a novel recursive Taylor expansion approach to analytically continue the zeta function and proves the Riemann Hypothesis unconditionally.

Findings

Non-trivial zeros are proven to lie on the critical line

The recursive Taylor expansion method simplifies the analysis of the zeta function

Contradiction arises from assuming off-line zeros, confirming their non-existence

Abstract

We present an unconditional proof that non-trivial zeros of the Riemann Zeta function must lie strictly on the critical line $\text{Re}(s) = 0.5$. By defining a recursive path of Taylor expansions originating from the domain of absolute convergence, we translate the zeta function towards the critical region, which is an easy-to-understand form of the analytical continuation. We then assume the existence of off-critical-line (off-line) zeros, which exist in pairs symmetric by the critical line. If the pairs are zero in value, their real and imaginary components differences should be both zero. However, we derive a contradiction against the assumption via basic logical deduction, proving the non-existence of the off-line zeros.