Date of Degree

9-2016

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Carlos J. Moreno

Committee Members

Brooke Feigon

Burton Randol

Subject Categories

Harmonic Analysis and Representation | Number Theory

Keywords

Weil explicit formula, Selberg trace formula, zeroes of L-functions

Abstract

In this thesis, motivated by an observation of D. Hejhal, we show that the explicit formulae of A. Weil for sums over zeroes of Hecke L-functions, via the Maass-Selberg relation, occur in the continuous spectral terms in the Selberg trace formula over various number fields. In Part I, we discuss the relevant parts of the trace formulae classically and adelically, developing the necessary representation theoretic background. In Part II, we show how show the explicit formulae intervene, using the classical formulation of Weil; then we recast this in terms of Weil distributions and the adelic formulation of Weil. As an application, we prove a lower bound for these explicit formulae using properties of the trace formula, in the spirit of Weil's criterion for the Riemann hypothesis.

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