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

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 Lindel\"of 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.