**Authors:** Carlos Castro

The Riemann's hypothesis (RH) states that the nontrivial zeros of the
Riemann zeta-function are of the form s_{n} = 1/2 + iλ_{n}. An improvement
of our previous construction to prove the RH is presented by implementing
the Hilbert-Polya proposal and furnishing the Fractal Supersymmetric
Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the
imaginary parts of the zeta zeros. We model the fractal fluctuations of the
smooth Wu-Sprung potential ( that capture the average level density of
zeros ) by recurring to P a weighted superposition of Weierstrass functions
ΣW(x,p,D) and where the summation has to be performed over all
primes p in order to recapture the connection between the distribution of
zeta zeros and prime numbers. We proceed next with the construction of
a smooth version of the fractal QM wave equation by writing an ordinary
Schroedinger equation whose fluctuating potential (relative to the smooth
Wu-Sprung potential) has the same functional form as the fluctuating part
of the level density of zeros. The second approach to prove the RH relies
on the existence of a continuous family of scaling-like operators involving
the Gauss-Jacobi theta series. An explicit completion relation ( "trace
formula") related to a superposition of eigenfunctions of these scaling-like
operators is defined. If the completion relation is satisfied this could be another
test of the Riemann Hypothesis. In an appendix we briefly describe
our recent findings showing why the Riemann Hypothesis is a consequence
of CT -invariant Quantum Mechanics, because < Ψ_{s} | CT | Ψ_{s} > ≠ 0
where s are the complex eigenvalues of the scaling-like operators.

**Comments:** 20 Pages. This article appeared in the Int. Jour. of Geom. Methods of Modern Physics, 4, no. 5 (2007) 881-895.

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[v1] 14 Sep 2009

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