... Several arithmetic and geometric information of $A$ can be deduced from the analytic properties of its zeta function $\zeta(A;s)=\sum_{m\in A}' q(m)^{-s}$; more precisely from its meromorphic continuation, the distribution of its poles, etc.. If $A$ has some algebraic or analytic regularity, one can then use the analytic or algebraic machinery to extend analytically $\zeta(A;s)$. ... The idea is to exploit self-similarity instead of algebraic or analytic regularity. Laboratoire Poncelet . ...