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arXiv · 2503.20455

Local average in the Hyperbolic sphere problem

Giacomo Cherubini·Christos Katsivelos
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Abstract

We consider a local average in the hyperbolic lattice point counting problem for the Picard group $\Gamma$ 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 $\Gamma$. We also use bounds on a spectral exponential sum over the Laplace eigenvalues for $\Gamma$, which has been studied in the context of the prime geodesic theorem and for which unconditional bounds are known.

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Giacomo Cherubini, Christos Katsivelos. 2025-03-26. Local average in the Hyperbolic sphere problem. https://arxiv.org/abs/2503.20455

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