The Prime Geodesic Theorem for the Picard Orbifold

TL;DR

This paper proves a prime geodesic theorem for the Picard orbifold, improving error estimates by leveraging advances in L-function bounds, zero density results, and Fourier coefficient analysis, with implications for longstanding conjectures.

Contribution

It establishes the prime geodesic theorem for the Picard orbifold with new explicit error bounds, unconditionally at 1.483 and conditionally at 1.425, using novel analytic techniques.

Findings

Unconditional exponent of 1.483 achieved.

Conditional exponent of 1.425 under GRH.

Enhanced understanding of prime geodesics in hyperbolic 3-orbifolds.

Abstract

We establish the prime geodesic theorem for the Picard orbifold $\mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathbb{H}^{3}$, wherein the error term shrinks proportionally to improvements in the subconvex exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Our result sheds light on a venerable conjecture by attaining an unconditional exponent of $1.483$ and a conditionally superior exponent of $1.425$ under the generalised Lindel\"{o}f hypothesis. The argument synthesises, among other elements, the complete resolution of Koyama's (2001) mean Lindel\"{o}f hypothesis over $\mathbb{Q}(i)$, an improved Brun-Titchmarsh-type theorem over short intervals, a bootstrapped multiplicative exponent pair in the limiting regime, and a zero density theorem for the symplectic family of quadratic characters. Notably, despite the theoretical strength of our manifestations towards the mean Lindel\"{o}f hypothesis, the fundamental toolbox relies exclusively on the optimal mean square asymptotics for the Fourier coefficients of Maass cusp forms via the pre-Kuznetsov formula.