On a prime zeta function of a graph (Q2256305)

Let \(X\) be a finite graph without vertices of degree \(1\). A prime in \(X\) is a cycle (closed path) that has no subcycles of length \(2\), and cannot be obtained from another cycle by walking around it several times. Cycles that differ only by the choice of the starting point are considered equivalent and define the same prime. The authors consider the prime zeta function of \(X\) defined as \[ P_X(u) = \sum_{P} u^{\ell(P)}, \] where \(p\) ranges over primes, and \(u\) is a complex variable with \(|u|\) sufficiently small. This function is analogous to the sum \(\sum_p p^{-s}\) over rational primes in the theory of the Riemann zeta function. Let \(R_X\) denote the radius of convergence of the graph zeta function \(Z_X(s)\) (and of the above series, cf. [\textit{A. Terras}, Zeta functions of graphs. A stroll through the garden. Cambridge: Cambridge University Press (2011; Zbl 1206.05003)]). The authors show that \[ P_X(s) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \log Z_X(u^n) \] for \(|u|<R_X\), where \(\mu(n)\) is the Möbius function. They show that \(P_X\) has an analytic continuation to a dense subset of the disk \(|u|<1\) and that the unit circle is the natural boundary for \(P_X(s)\). They also prove graph-theoretic analogues of Mertens' first, second and third theorems.