  We study the behavior of the Riemann zeta function on the critical line when
the imaginary part of the argument is sampled by the Cauchy random walk. We
develop a complete second order theory for the corresponding system of random
variables and show that it behaves almost like a system of non-correlated
variables. Exploiting this fact in relation with known criteria for almost sure
convergence allows to investigate its almost sure asymptotic behavior.
