On Zeta functions and $\mu$-series of string algebras

Let $\overline\mu_\Lambda(t):=\sum\limits_{m\geq1}\mu_\Lambda(m)t^m$ be the \emph{$\mu$-series} of a finite-dimensional tame algebra $\Lambda$ over an algebraically closed field, where $\mu_\Lambda(m)$ denotes the minimal number of one-parameter families of $\Lambda$-modules with total dimension $m$. When $\Lambda$ is a string algebra with $\mathrm{Ba}(\Lambda)$ as its set of bands up to cyclic permutation, define the \emph{zeta function} $\zeta_\Lambda(t):=\prod\limits_{\mathfrak b\in\mathrm{Ba}(\Lambda)}(1-t^{|\mathfrak b|})^{-1}$, where $|\mathfrak b|$ is the length of $\mathfrak b$. We prove an analogue of the prime number theorem for string algebras and use it to conclude that non-domestic string algebras are of exponential growth. Finally, we show that a string algebra is domestic if and only if its $\mu$-series is rational.