## 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 ρ_{n} (imaginary parts) of the zeta zeros in the critical line :
s_{n} = 1/2 + iρ_{n}
A detailed analysis of a one-dimensional Dirac-like operator
with a potential V(x) is given that reproduces the spectrum of energy levels
E_{n} = ρ_{n}, when the boundary conditions
Ψ_{E} (x = -∞) = ± Ψ_{E} (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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### Submission history

[v1] 21 Aug 2009

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