Second Moment of the Prime Geodesic Theorem for $\mathrm{PSL}(2,\mathbb{Z}[i])$

Abstract

The remainder $E_{\Gamma }(X)$ in the Prime Geodesic Theorem for the Picard group $\Gamma =\mathrm{PSL}(2,\mathbb{Z}[i])$ is known to be bounded by $O(X^{3/2+\epsilon })$ under the assumption of the Lindelöf hypothesis for quadratic Dirichlet $L$-functions over Gaussian integers. By studying the second moment of $E_{\Gamma }(X)$, we show that on average the same bound holds unconditionally.

13 pages