  The validity of the Riemann Hypothesis (RH) on the location of the
non-trivial zeros of the Riemann zeta-function is directly related to the
growth of the Mertens function: the RH is indeed true if the Mertens function
goes asymptotically as $M(x) \simeq x^{1/2 + \epsilon}$. We show that this
behavior can be established on the basis of a new probabilistic approach based
on the global properties of the Mertens function. To this aim, we focus the
attention on the square-free numbers and we derive a series of probabilistic
results concerning the prime number distribution along the series of
square-free numbers, the average number of prime divisors, the Erdos-Kac
theorem for square-free numbers, etc. These results lead us to the conclusion
that the Mertens function is subject to a normal distribution as much as any
other random walk, therefore with an asymptotic behaviour given by $x^{1/2
+\epsilon}$. We also argue how the Riemann Hypothesis implies the Generalised
Riemann Hypothesis for the Dirichlet $L$-functions. Next we study the local
properties of the Mertens function dictated by the Mobius coefficients
restricted to the square-free numbers. We perform a massive statistical
analysis on these coefficients, applying to them a series of randomness tests
of increasing precision and complexity. The successful outputs of all these
tests (with a level of confidence of 99% that all the sub-sequences analyzed
are indeed random) can be seen as impressive experimental confirmations of the
probabilistic normal law distribution of the Mertens function analytically
established earlier. In view of the theoretical probabilistic argument and the
large battery of statistical tests, we can conclude that while a violation of
the RH is strictly speaking not impossible, it is however ridiculously
improbable.
