Description
This talk presents two interconnected spectral and geometric results on compact Riemann surfaces equipped with unitary multiplier systems.
First, we establish an explicit asymptotic formula for the prime geodesic theorem on a fixed compact Riemann surface $X$ of genus $g$ with a unitary multiplier system $\chi$. We analyze the weighted Chebyshev-like function
$$\Psi(x,\chi) = \sum_{P\in\Gamma,\, \text{Tr}(P) > 2, \, N(P)\leq x} \Lambda(P)\text{Tr}(\chi(P))$$
and deduce an asymptotic distribution for large $x$ with an explicit remainder bound in which the implied constant depends on the dimension of the multiplier system, the genus, the systole, the spectral gap, and the number of small eigenvalues.
Second, we investigate the asymptotic behavior of the regularized determinant for a sequence of compact Riemann surfaces $X_n$ as the genus tends to infinity. Under a weak spectral gap assumption and a uniform lower bound on the systole, we study the normalized log-determinant of the sequence of weighted Laplacians $\Delta_{n,\chi_n, 2k_n}$ with $\alpha = \lim k_n \in [0,1]$. Assuming a mild geometric hypothesis on the density of short geodesics, we prove that the normalized log-determinant converges to a universal constant as $n \to \infty$. This limiting value is given explicitly as a sum of an absolute constant, the Gamma function, and the Barnes $G$-function depending on the limiting weight $\alpha$.
The work presented is joint with Jay Jorgenson and Polyxeni Spilioti.