Prime geodesic theorem for the Picard manifold

Abstract

Let Γ=PSL(2,Z[i]) be the Picard group and H3 be the three-dimensional hyperbolic space. We study the Prime Geodesic Theorem for the quotient Γ∖H3, called the Picard manifold, obtaining an error term of size O(X3/2+θ/2+ϵ), where θ denotes a subconvexity exponent for quadratic Dirichlet L-functions defined over Gaussian integers.