  The universality classes of the quantum Hall transitions are considered in
terms of fractal sets of dual topological quantum numbers filling factors,
labelled by a fractal or Hausdorff dimension defined into the interval $1 < h <
2$ and associated with fractal curves. We show that our approach to the
fractional quantum Hall effect-FQHE is free of any empirical formula and this
characteristic appears as a crucial insight for our understanding of the FQHE.
According to our formulation, the FQHE gets a fractal structure from the
connection between the filling factors and the Hausdoff dimension of the
quantum paths of particles termed fractons which obey a fractal distribution
function associated with a fractal von Neumann entropy. This way, the quantum
Hall transitions satisfy some properties related to the Farey sequences of
rational numbers and so our theoretical description of the FQHE establishes a
connection between physics, fractal geometry and number theory. The FQHE as a
convenient physical system for a possible prove of the Riemann hypothesis is
suggested.
