Prime geodesic theorem for the Picard manifold

Abstract

Let $\Gamma =PSL(2,Z[i])$ be the Picard group and $H^3$ be the three-dimensional hyperbolic space. We study the Prime Geodesic Theorem for the quotient $\Gamma \setminus H^3$, called the Picard manifold, obtaining an error term of size $O(X^{3/2+\theta /2+\epsilon })$, where $\theta$ denotes a subconvexity exponent for quadratic Dirichlet $L$-functions defined over Gaussian integers.

Corrected subconvexity estimate in (3.24)