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 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 202139+ in the prime geodesic theorem under the generalised Lindel\"of hypothesis = 0, improving upon the existing conditional results. This provides further solid evidence towards the validity of the conjectural exponent 1+.