A note on the possibility of proving the Riemann hypothesis

TL;DR

This paper explores the Riemann Hypothesis by assuming a counterexample and demonstrating, through algebraic analysis, that such a counterexample leads to a contradiction, thus supporting the hypothesis.

Contribution

It presents a novel reductio ad absurdum approach using the Gram-Backlund formula and algebraic constraints to argue for the truth of the Riemann Hypothesis.

Findings

Assuming a non-critical zero leads to an algebraic contradiction.

The analysis shows zeroes off the critical line cannot satisfy the derived constraints.

Supports the Riemann Hypothesis by contradiction.

Abstract

As well known, the important hypothesis formulated by B.G. RIEMANN in 1859 states that all non-trivial zeroes of the Zeta function $Z(s)=\sum_{n=1}^{\infty } n^{-s}$ should fall on the Critical Line (C.L.) $Re(s)=\frac{1}{2}$.\\ Although direct numerical search of the zeroes failed to identify any outlier, i.e. any zeroes with $Re(s)\neq\frac{1}{2}$, a general proof of the Hypothesis has not yet been found.\\ The present Note aims to approach the problem from a 'reductio ad absurdum' way, i.e. it assumes that an outlier pair of c.c. zero-points, $s=\frac{1}{2}+\xi \pm i.Y_H$ with $\xi\neq0$, has been found, and then proceeds to analyze what are the implications of this assumption. Starting from the well-known GRAM-BACKLUND formula for an explicit expression of the Zeta function, the Fundamental Theorem of Algebra (FTA) allows to evidence, through legitimate algebraic manipulations, the necessity that the assumed outlying pair of c.c. zero-points fulfils an implicit additional constraint. The zero-condition according to the GRAM- BACKLUND formulation and this additional constraint are seen to be mutually incompatible unless the pair of c.c. zero-points belong to the C.L.. This conclusion is equivalent to a verification of the RIEMANN Hypothesis.