  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 $M(x) \,=\,\sum_{k=1}^x \mu(k)$, where $\mu(k)$
is the M\"{o}bius coefficient of the integer $k$: the RH is indeed true if the
Mertens function goes asymptotically as $M(x) \sim x^{1/2 + \epsilon}$, where
$\epsilon$ is an arbitrary strictly positive quantity. This behavior can be
established on the basis of a new probabilistic approach based on the global
properties of Mertens function. To this aim we derive a series of probabilistic
results concerning the prime number distribution along the series of
square-free numbers which shows that the Mertens function is subject to a
normal distribution. We also show that the validity of the RH also implies the
validity of the Generalized Riemann Hypothesis for the Dirichlet $L$-functions.
Next we study the local properties of the Mertens function, i.e. its variation
induced by each M\"{o}bius coefficient 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, for a
total number of eighteen different tests. The successful outputs of all these
tests (each of them 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 brownian nature of the restricted
M\"{o}bius coefficients and 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 extremely improbable.
