The Prime Geodesic Theorem for the Picard Manifold

We prove the prime geodesic theorem for the Picard manifold whose error term shrinks as the hybrid subconvex exponent θ for quadratic Dirichlet L-functions over Gaussian integers decreases. Our rate of decay is faster than that of Balkanova-Frolenkov (2022) and Kaneko (2022), and leads to the exponent 202/139+ε in the prime geodesic theorem under the generalised Lindel\"{o}f hypothesis θ = 0, improving upon the existing conditional results. This provides further solid evidence towards the validity of the conjectural exponent 1+ε.