| 1. |
- Balkanova, Olga, 1988, et al.
(författare)
-
Prime geodesic theorem in the 3-dimensional hyperbolic space
- 2019
-
Ingår i: Transactions of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9947 .- 1088-6850. ; 372:8, s. 5355-5374
-
Tidskriftsartikel (refereegranskat)abstract
- For Γ a cofinite Kleinian group acting on H3, we study the prime geodesic theorem on M = Γ\H3, which asks about the asymptotic behavior of lengths of primitive closed geodesics (prime geodesics) on M. Let EΓ(X) be the error in the counting of prime geodesics with length at most log X. For the Picard manifold, Γ = PSL(2, Z[i]), we improve the classical bound of Sarnak, EΓ(X) = O(X5/3+e), to EΓ(X) = O(X13/8+e). In the process we obtain a mean subconvexity estimate for the Rankin-Selberg L-function attached to Maass-Hecke cusp forms. We also investigate the second moment of EΓ(X) for a general cofinite group Γ, and we show that it is bounded by O(X16/5+e).
|
|
| 2. |
- Laaksonen, Niko
(författare)
-
Lattice point counting in sectors of Hyperbolic 3-space
- 2017
-
Ingår i: Quarterly Journal of Mathematics. - : Oxford University Press. - 0033-5606 .- 1464-3847. ; 68:3, s. 891-922
-
Tidskriftsartikel (refereegranskat)abstract
- Let G be a cocompact discrete subgroup of PSL2(.) and denote by. the three-dimensional upper half-space. For a p I., we count the number of points in the orbit Gp, according to their distance, arccosh X, from a totally geodesic hyperplane. The main term in n dimensions was obtained by Herrmann for any subset of a totally geodesic submanifold. We prove a pointwise error term ofO(X3 2) by extending the method of Huber and Chatzakos-Petridis to three dimensions. By applying Chamizo's large sieve inequalities, we obtain the conjectured error term O(X1+ e) on an average in the spatial aspect. We prove a corresponding large sieve inequality for the radial average and explain why it only improves on the pointwise bound by 1/ 6.
|
|
| 3. |
- Laaksonen, Niko, et al.
(författare)
-
ON THE VALUE DISTRIBUTION OF TWO DIRICHLET L-FUNCTIONS
- 2018
-
Ingår i: FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI. - : WYDAWNICTWO NAUKOWE UAM. - 0208-6573. ; 58:1, s. 43-68
-
Tidskriftsartikel (refereegranskat)abstract
- Let rho denote the non-trivial zeros of the Riemann zeta function. We study the relative value distribution of L(rho + sigma, chi(1)) and L(rho + sigma, chi(2)), where sigma is an element of[0, 1/2) is fixed and chi(1), chi(2) are two fixed Dirichlet characters to distinct prime moduli. For sigma > 0 we prove that a positive proportion of these pairs of values are linearly independent over R, which implies that the arguments of the values are different. For sigma = 0 we show that, up to height T, the values are different for cT of the Riemann zeros for some positive constant c.
|
|
