arXiv subjects

Christos Katsivelos

Publications and source records attributed to Christos Katsivelos.

2 recordsLinked to original sources

arXiv2026

Local average in the Hyperbolic sphere problem

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

math.NT↗

arXiv2025

The hyperbolic lattice counting problem in large dimensions

For $n\geq 3$ and $Γ$ a cocompact lattice acting on the hyperbolic space $\mathbb{H}^n$, we investigate the average behaviour of the error term in the circle problem. First, we explore the local average of the error term over compact sets of $Γ\backslash\mathbb{H}^n$. Our upper bound depends on the quantum variance and the spectral exponential sums appearing in the study of the Prime geodesic theorem. We also prove $Ω$-results for the mean value and the second moment of the error term.

math.NT↗