Final steps towards a proof of the Riemann hypothesis

TL;DR

This paper claims to present a proof of the Riemann hypothesis by constructing operators and eigenfunctions whose orthogonality relates directly to the zeros of the zeta function, leveraging symmetries and fundamental relations.

Contribution

It introduces a novel operator framework and symmetry-based approach that links eigenfunction orthogonality to the zeros of the Riemann zeta function, aiming to prove RH.

Findings

Establishes a one-to-one correspondence between eigenstates and zeta zeros.

Shows that excluding zeros off the critical line leads to zeros only at 1/2 + iλ.

Connects symmetries in variables to the distribution of zeta zeros.

Abstract

A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective eigenfunctions \psi_s (t), parameterized by continuous real indexes k and l. Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function. Due to the fundamental Gauss-Jacobi relation and the Riemann fundamental relation Z (s') = Z (1-s'), one can show that there is a direct concatenation among the following symmetries, t goes to 1/t, s goes to \beta - s (\beta a real), and s' goes to 1 - s', which establishes a one-to-one correspondence between the label s of one orthogonal state to a unique vacuum state, and a zero s' of the \zeta. It is shown that the RH is a direct consequence of these symmetries, by arguing in particular that an exclusion of a continuum of the zeros of the Riemann zeta function results in the discrete set of the zeros located at the points s_n = 1/2 + i \lambda_n in the complex plane.