Randomness of Mobius coefficents and brownian motion: growth of the Mertens function and the Riemann Hypothesis

Abstract

The validity of the Riemann Hypothesis (RH) on the location of the non-trivial zeros of the Riemann $ζ$-function is directly related to the growth of the Mertens function $M(x) \,=\,\sum_{k=1}^x μ(k)$, where $μ(k)$ is the Möbius coefficient of the integer $k$: the RH is indeed true if the Mertens function goes asymptotically as $M(x) \sim x^{1/2 + ε}$, where $ε$ 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 Möbius 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 Möbius 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.