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

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 log x )) 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.