The spectral concentration for damped waves on compact Anosov manifolds

We study the spectral distribution of damped waves on compact Anosov manifolds. Sj\"ostrand \cite{SJ1} proved that the imaginary parts of the majority of the eigenvalues concentrate near the average of the damping function, see also Anantharaman \cite{AN2}. In this paper, we prove that the most of eigenvalues actually lie in certain regions with imaginary parts that approach the average logarithmically as the real parts tend to infinity. The proof relies on the moderate deviation principles for Anosov geodesic flows. As an application, we show the concentration of non-trivial zeros of twisted Selberg zeta functions in a logarithmic region asymptotically close to $\Re s=\frac{1}{2}$.