Local average in the hyperbolic sphere problem
Abstract
We consider a local average in the hyperbolic lattice point counting problem for the Picard group Γ \Gamma acting on the three-dimensional hyperbolic space. Compared to the pointwise case, we improve the bounds on the remainder in the counting, conditionally on a quantum variance estimate for Maass cusp forms attached to Γ \Gamma . We also use bounds on a spectral exponential sum over the Laplace eigenvalues for Γ \Gamma , which has been studied in the context of the prime geodesic theorem and for which unconditional bounds are known.
Fields of Study & Subjects
Analytic Number Theory Research
Advanced Algebra and Geometry
Spectral Theory in Mathematical Physics
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 3
- Pages
- 1041–1056
- Publication date
- 14 January 2026
- Language
- EN