The low activity phase of some Dirichlet series

We show that a rigorous statistical mechanics description of some Dirichlet series is possible. Using the abstract polymer model language of statistical mechanics and the polymer expansion theory we characterize the low activity phase by the suitable exponential decay of the truncated correlation functions. The result is obtained with a finite-volume approximation of the Dirichlet series which turns out to be the gran canonical partition function of a hard-core interacting system. The correlation functions, which have a deep number theoretical meaning being the probability of suitable divisibility properties, are controlled, in the thermodynamical limit, by means of the Kirkwood-Salsburg type iterative equations.