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Distribution results in automorphic forms and analytic number theory

By Jimi Lee Truelsen
PhD Dissertations
August 2009

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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