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

Abstract

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