2002/08/30 by Carlos Castro, Castro, Carlos, Jorge Mahecha +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #hep-th #math-ph #math.GM #math.MP
paper · pdf · doi:10.48550/arxiv.hep-th/0208221
Latex file, 18 pages, revised text, stronger and improved arguments with a figure are added, submitted to Annals of Mathematics
arxiv created 2002/11/05 · arxiv updated 2009/11/30
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 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 (βa 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 + i λn in the complex plane.