Number Theory


On the Riemann Hypothesis, Area Quantization, Dirac Operators, Modularity and Renormalization Group

Authors: Carlos Castro

Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert-Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line : sn = 1/2 + iρn A detailed analysis of a one-dimensional Dirac-like operator with a potential V(x) is given that reproduces the spectrum of energy levels En = ρn, when the boundary conditions ΨE (x = -∞) = ± ΨE (x = +∞) are imposed. Such potential V(x) is derived implicitly from the relation x = x(V) = π/2(dN(V)/dV), where the functional form of N(V) is given by the full-fledged Riemann-von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the O(E-n) terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr-Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large x (O(1/logx)) has a one-to-one correspondence with the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the Renormalization Group program.

Comments: 33 pages, This article will appear in the Int. J. of Geom. Methods in Mod Phys vol 7, no. 1 (2010)

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[v1] 21 Aug 2009

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