## 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)

**Download:** **PDF**

### Submission history

[v1] 21 Aug 2009

**Unique-IP document downloads:** 459 times

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*