Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{\mathrm{\Gamma },\mathrm{\Sigma }}^{\mathrm{o}}(\lambda )$ associated to the spinor bundle $\mathrm{\Sigma }$ on odd dimensional convex co-compact hyperbolic manifolds $X:=\mathrm{\Gamma }\mathrm{\backslash }{\text{hh}}^{2n+1}$. We define a natural eta invariant $\eta (D)$ associated to the Dirac operator $D$ on $X$ and prove that $\eta (D)=\frac{1}{\pi i}\log Z_{\mathrm{\Gamma },\mathrm{\Sigma }}^{\mathrm{o}}(0)$, thus extending Millson’s formula to this setting. As a byproduct, we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We also define an eta invariant for the odd signature operator and, under some conditions, we describe it on the Schottky space of 3-dimensional Schottky hyperbolic manifolds and relate it to Zograf factorization formula.