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.