#### Date of Award

Spring 5-13-2017

#### Level of Access

Campus-Only Thesis

#### Degree Name

Master of Arts (MA)

#### Department

Mathematics

#### Advisor

Andrew Knightly

#### Second Committee Member

Nigel Pitt

#### Third Committee Member

Yannan Qiu

#### Abstract

If two bodies are orbiting in the same plane at different velocities, where do they align? We will discuss the history of this problem and consider the sequence of points where bodies align. In particular, we will look at the density of this sequence on the circle with unit circumference, the size of the gaps between successive points, and the uniformity of these gaps. This will lead to a discussion of what it means for a sequence to be uniformly distributed modulo 1, a concept which is stronger than density. This is all covered in Chapter 1. We will define weighted uniform distribution and uniform distribution relative to measures other than the Lebesgue measure.

Chapter 2 will define a specific set of orthonormal polynomials, the Chebyshev polynomials of the second kind. These polynomials will be useful to prove the main result of the thesis, which is about the distribution of certain sequences relative to a measure on [-2,2] called the Sato-Tate measure. Chapter 3 will give us the background needed about modular forms along with examples and results about certain types of modular forms. Lastly, Chapter 4 will bring together results from all the previous chapters. We will give three results about uniform distribution of Hecke eigenvalues of modular forms. The first two are known results while the last is a new result.

#### Recommended Citation

Reno, Caroline, "Uniform Distribution of Hecke Eigenvalues" (2017). *Electronic Theses and Dissertations*. 2688.

https://digitalcommons.library.umaine.edu/etd/2688