#### Abstract

We study the analytic properties of the Eisenstein Series of $frac {1}{2}$-integral weight associated with the Hecke congruence subgroup $Gamma_0(4)$. Using these properties we obtain asymptotics for sums of certain Dirichlet $L$-series. We also obtain a formula reducing the study of Selberg's Eigenvalue Conjecture to the study of the nonvanishing of the Eisenstein Series $E(z,s)$ for Hecke congruence subgroups $Gamma_0(N)$ at $s=frac {1+i}{2}$.

#### Degree

MS

#### College and Department

Physical and Mathematical Sciences; Mathematics

#### Rights

http://lib.byu.edu/about/copyright/

#### BYU ScholarsArchive Citation

Belt, Dustin David, "Topics on the Spectral Theory of Automorphic Forms" (2006). *All Theses and Dissertations*. 490.

http://scholarsarchive.byu.edu/etd/490

#### Date Submitted

2006-07-12

#### Document Type

Thesis

#### Handle

http://hdl.lib.byu.edu/1877/etd1423

#### Keywords

automorphic forms, Eisenstein series, Dirichlet series, Selberg's Eigenvalue Conjecture