# Distribution results in automorphic forms and analytic number theory

By Jimi Lee Truelsen
PhD Dissertations
August 2009
Abstract:

In this thesis three different distribution problems are studied. In the first problem we consider the Eisenstein series $E(g,s,\chi)$ on $GL(2,A)$, where $A$ is the adele ring of a number field. We prove (quantitatively) that the measure $| E(g,1/2+it,\chi)|^2 d\mu$ becomes equidistributed in the limit $t \to \infty$. Here $d\mu$ is the measure derived from the Haar measure on $GL(2,A)$. This generalizes previous results due to W. Luo and P. Sarnak and S. Koyama.

The second problem concerns angles in hyperbolic lattices. We prove that in a suitable (and natural) setting these angles are equidistributed with an effective error term for the equidistribution rate. We use this to generalize a result due to F. Boca.

The last problem studied in the thesis is about the pair correlation for the fractional parts of $n^2\alpha$. It has been proved by Z. Rudnick and P. Sarnak that the pair correlation is Poissonian for almost all $\alpha$. However, one does not know of any specific $\alpha$ for which it holds. We show that the problem is closely related to a divisor problem, which gives a better arithmetic understanding of the problem. The divisor problem considered seems to be hard, but we can show that it is true on average in a suitable sense.

Thesis advisors: Morten S. Risager and Alexei B. Venkov
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