Complex dimensions for IFS with overlaps

The notion of complex dimension of a one-dimensional Cantor set $C=\bigcap_{n=1}^\infty C_n$ dates back decades. It is defined as the set of poles of the meromorphic $\zeta$-function $\zeta(s)=\sum_{n=1}^{\infty}d_j^s$, where $\Re s>0$, and $d_j$ is the length of the $j$th interval in $C_n$. Following the trend, I switch from sets to measures, which will allow me to generalize the construction to iterated function schemes that do not necessarily satisfy the Open Set Condition.