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 Teichmuller space. We then use this formula to determine the asymptotic behavior as $text{Re} (s) to infty$ 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 $H^0(X_t, mathcal K_t)to tin mathcal T$ of holomorphic m-differentials over the Teichmuller 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.