Local average in hyperbolic lattice point counting, with an Appendix by Niko Laaksonen

Abstract

The hyperbolic lattice point problem asks to estimate the size of the orbit Γz inside a hyperbolic disk of radius cosh −1 (X/2) for Γ a discrete subgroup of PSL2(R). Selberg proved the estimate O(X 2/3 ) for the error term for cofinite or cocompact groups. This has not been improved for any group and any center. In this paper local averaging over the center is investigated for PSL2(Z). The result is that the error term can be improved to O(X 7/12+ε ). The proof uses surprisingly strong input e.g. results on the quantum ergodicity of Maaß cusp forms and estimates on spectral exponential sums. We also prove omega results for this averaging, consistent with the conjectural best error bound O(X 1/2+ε ). In the appendix the relevant exponential sum over the spectral parameters is investigated.

Mean square in the prime geodesic theorem

“…In the appendix to [21] Laaksonen provides numerics that support this conjecture. In the proof of Theorem 1.4 (see Proposition 4.5) we prove that we have…”

Bounds for a spectral exponential sum

“…Petridis and Risager [, Conjecture 2.2] conjectured square root cancellation in sum , namely Furthermore, they showed that estimate yields not only the best possible error term in the prime geodesic theorem, but also the best error term on average for the hyperbolic lattice problem. See for more details.…”

“…In [, Appendix], Laaksonen proved that the conjecture of Petridis and Risager is true for a fixed as . Moreover, for where is the distance from to the nearest integer, Laaksonen mentioned in [, Experimental Observation 2] that has a peak when .…”

CM-points and lattice counting on arithmetic compact Riemann surfaces

“…More information can be obtained for the error term when the group Γ is arithmetic. For Γ = PSL 2 (Z), Petridis and Risager recently studied the average growth of E(X; z, z) on compact subsets of the modular surface Γ\H both in the local [31] and the discrete aspect [32]. To state their discrete average result, for d < 0 a fundamental discriminant denote by Λ d the set of CM (or Heegner) points of discriminant d and by h(d) the class number h(d) = #Λ d .…”