The Prime Geodesic Theorem and Bounds for Character Sums

We establish the prime geodesic theorem for the modular surface with exponent 113164+, improving upon the long-standing exponent 2536+ of Soundararajan-Young (2013). Our argument goes through a well-trodden trail via the automorphic machinery, and refines the methods of Iwaniec (1984) and Cai (2002) to a maximum extent. Furthermore, we apply Burgess bounds for character sums in short intervals, combined with hybrid Weyl-strength subconvex bounds for quadratic Dirichlet L-functions due to Petrow-Young (2023).