Final steps towards a proof of the Riemann hypothesis

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 in one dimension, and their respective eigenfunctions ψs(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 β − s (β 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 ζ. 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 sn = 1/2+ iλn in the complex plane.