Second variation of Selberg zeta functions and curvature asymptotics

We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichm"uller space. We then use this formula to determine the asymptotic behavior as $\text{Re}(s)\to \mathrm{\infty }$ of the second variation. As a consequence, for $m\in \mathbb{N}$, we obtain the complete expansion in $m$ of the curvature of the vector bundle $H^0(X_t, \mathcal K_t)\to t\in \mathcal T$ of holomorphic m-differentials over the Teichm"uller space $\mathcal{T}$, for $m$ large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $O(m^2{e}^{-l_{0}m}),$ where $l_0$ is the length of the shortest closed hyperbolic geodesic.