The M bius function and statistical mechanics

We consider a probabilistic model for square-free numbers, and provide limit theorems for several random variables defined in our ensemble. The limit transition corresponds to the thermodynamical limit in Statistical Mechanics. We also prove some inequalities inspired by a recent conjecture by P. Sarnak concerning the randomness in the M bius sequence, and discuss a method of summation for the Riemann zeta function ζ(s) on the vertical line ${\mathfrak{R}s = 1}$ .