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

Abstract

Let ${\overline{\mu }}_{\Lambda }(t):=\sum _{m\ge 1}{\mu }_{\Lambda }(m)t^m$ 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} ${\zeta }_{\Lambda }(t):=\prod _{\mathfrak{b}\in \mathrm{Ba}(\Lambda )}(1-t^{|\mathfrak{b}|}{)}^{-1}$, 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.

12 pages