Prime geodesic theorem for the Picard manifold

TL;DR

This paper investigates the distribution of prime geodesics on the Picard manifold, providing an improved error term in the Prime Geodesic Theorem linked to subconvexity bounds for quadratic Dirichlet L-functions over Gaussian integers.

Contribution

It establishes a new error term for the Prime Geodesic Theorem on the Picard manifold, connecting geometric properties with analytic number theory.

Findings

Derived an error term of size O(X^{3/2+θ/2+ε}) for the Prime Geodesic Theorem.

Linked the error term to subconvexity bounds of quadratic Dirichlet L-functions over Gaussian integers.

Enhanced understanding of prime geodesic distribution in hyperbolic 3-manifolds.

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.