On the functional equation of twisted Ruelle zeta function and Fried's conjecture
Let $M$ be a finite volume hyperbolic Riemann surface with arbitrary signature, and let $\chi$ be an arbitrary $m$-dimensional multiplier system of weight $k$. Let $R(s,\chi)$ be the associated Ruelle zeta function, and $\varphi(s,\chi)$ the determinant of the scattering matrix. We prove the functional equation that $R(s,\chi)\varphi(s,\chi) = R(-s,\chi)\varphi(s,\chi)H(s,\chi)$ where $H(s,\chi)$ is a meromorphic function of order one explicitly determined using the topological data of $M$ and of $\chi$, and the trigonometric function $\sin(s)$. From this, we determine the order of the divisor of $R(s,\chi)$ at $s=0$ and compute the lead coefficient in its Laurent expansion at $s=0$. When combined with results by Kitano and by Yamaguchi, we prove further instances of the Fried conjecture, which states that the R-torsion of the above data is simply expressed in terms of $R(0,\chi)$.