Prime Geodesic Theorem in the 3-dimensional Hyperbolic Space

For $\Gamma$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=\Gamma \backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics) on $M$. Let $E_{\Gamma}(X)$ be the error in the counting of prime geodesics with length at most $\log X$. For the Picard manifold, $\Gamma=\mathrm{PSL}(2,\mathbb{Z}[i])$, we improve the classical bound of Sarnak, $E_{\Gamma}(X)=O(X^{5/3+\epsilon})$, to $E_{\Gamma}(X)=O(X^{13/8+\epsilon})$. 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 $E_{\Gamma}(X)$ for a general cofinite group $\Gamma$, and show that it is bounded by $O(X^{16/5+\epsilon})$.Comment: Corrected proof of Theorem 3.3 (with a weaker bound), added two authors, 18 page

Lattice point problems in the hyperbolic plane

In this thesis we investigate two different lattice point problems in the hyperbolic plane, the classical hyperbolic lattice point problem and the hyperbolic lattice point problem in conjugacy classes. In order to study these problems we use tools from the harmonic analysis on the hyperbolic plane H

Distribution of Angles in Hyperbolic Lattices

We prove an effective equidistribution result about angles in a hyperbolic lattice. We use this to generalize a result due to F. P. Boca.Comment: 13 pages, 3 figures, Lots of minor correction

The prime geodesic theorem for PSL2(Z[i])\mathrm{PSL}_{2}(\mathbb{Z}[i]) and spectral exponential sums

We shall ponder the Prime Geodesic Theorem for the Picard manifold $\mathcal{M} = \mathrm{PSL}_{2}(\mathbb{Z}[i]) \backslash \mathfrak{h}^{3}$, which asks about the asymptotic behaviour of a counting function for the closed geodesics on $\mathcal{M}$. Let $E_{\Gamma}(X)$ be the error term arising from counting prime geodesics, we then prove the bound $E_{\Gamma}(X) \ll X^{3/2+\epsilon}$ on average, as well as various versions of pointwise bounds. The second moment bound is the pure counterpart of work of Balog et al. for $\Gamma = \mathrm{PSL}_{2}(\mathbb{Z})$, and the main innovation entails the delicate analysis of sums of Kloosterman sums with an explicit evaluation of oscillatory integrals. Our pointwise bounds concern Weyl-type subconvex bounds for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. Interestingly, we are also able to establish an asymptotic law for the spectral exponential sum in the spectral aspect for a cofinite Kleinian group $\Gamma$. Finally, we produce numerical experiments of its behaviour, visualising that $E_{\Gamma}(X)$ obeys a conjectural bound of the size $O(X^{1+\epsilon})$.Comment: Numerous improvements to the exposition; improved the quality of the main theorem (Theorem 1.1) and achieved additional theorems such as Theorems 1.4, 3.17, 4.1, and 5.

Quantum limits, counting and Landau-type formulae in hyperbolic space

In this thesis we explore a variety of topics in analytic number theory and automorphic forms. In the classical context, we look at the value distribution of two Dirichlet L-functions in the critical strip and prove that for a positive proportion these values are linearly independent over the real numbers. The main ingredient is the application of Landau's formula with Gonek's error term. The remainder of the thesis focuses on automorphic forms and their spectral theory. In this setting we explore three directions. First, we prove a Landau-type formula for an exponential sum over the eigenvalues of the Laplacian in PSL(2, Z)\H by using the Selberg Trace Formula. Next, we look at lattice point problems in three dimensions, namely, the number of points within a given distance from a totally geodesic hyperplane. We prove that the error term in this problem is O(X^{3/2}), where arccosh(X) is the hyperbolic distance to the hyperplane. An application of large sieve inequalities provides averages for the error term in the radial and spatial aspect. In particular, the spatial average is consistent with the conjecture that the pointwise error term is O(X^{1+\epsilon}). The radial average is an improvement on the pointwise bound by 1/6. Finally, we identify the quantum limit of scattering states for Bianchi groups of class number one. This follows as a consequence of studying the Quantum Unique Ergodicity of Eisenstein series at complex energies

Counting arcs in negative curvature

Let M be a complete Riemannian manifold with negative curvature, and let C_-, C_+ be two properly immersed closed convex subsets of M. We survey the asymptotic behaviour of the number of common perpendiculars of length at most s from C_- to C_+, giving error terms and counting with weights, starting from the work of Huber, Herrmann, Margulis and ending with the works of the authors. We describe the relationship with counting problems in circle packings of Kontorovich, Oh, Shah. We survey the tools used to obtain the precise asymptotics (Bowen-Margulis and Gibbs measures, skinning measures). We describe several arithmetic applications, in particular the ones by the authors on the asymptotics of the number of representations of integers by binary quadratic, Hermitian or Hamiltonian forms.Comment: Revised version, 44 page