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

Laaksonen, NPJ; (2016) Quantum limits, counting and Landau-type formulae in hyperbolic space. Doctoral thesis , UCL (University College London). Green open access

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Abstract

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.

Type: Thesis (Doctoral)
Title: Quantum limits, counting and Landau-type formulae in hyperbolic space
Event: UCL (University College London)
Open access status: An open access version is available from UCL Discovery
Language: English
Keywords: automorphic forms, L functions, Riemann zeta function, analytic number theory
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: https://discovery.ucl.ac.uk/id/eprint/1474273
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