#### Document Type

Dissertation

#### Date of Degree

Spring 2016

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Yangbo Ye

#### Abstract

The primary arithmetic information attached to a Maass cusp form is its Laplace eigenvalue. However, in the case of cuspidal Maass forms, the range that these eigenvalues can take is not well-understood. In particular it is unknown if, given a real number r, one can prove that there exists a primitive Maass cusp form with Laplace eigenvalue 1/4 + r^{2}. Conversely, given the Fourier coeﬃcients of a primitive Maass cusp form f on Γ_{0}(D), it is not clear whether or not one can determine its Laplace eigenvalue. In this paper we show that given only a ﬁnite number of Fourier coeﬃcients one can ﬁrst determine the level D, and then compute the Laplace eigenvalue to arbitrarily high precision. The key to our results will be understanding the resonance and rapid decay properties of Maass cusp forms. Let *f* be a primitive Maass cusp form with Fourier coeﬃcients λ*f* (n). The resonance sum for *f* is given by S_{X}(*f*;α;β) = Ε_{n≥1}λ*f*(n)‑Φ(^{n}/_{X}) e(αn^{β}) where φ ∈ C_{c}∞((1, 2)) is a Schwartz function and α ∈ R and β, X > 0 are real numbers. Sums of this form have been studied for many diﬀerent classes of functions *f*, including holomorphic modular forms for SL(2, Z), and Maass cusp forms for SL(n,Z). In this paper we take *f* to be a primitive Maass cusp form for a congruence subgroup Γ_{0}(D) ⊂ SL(2, Z). Thus our result extends the family of automorphic forms for which their resonance properties are understood. Similar analysis and algorithms can be easily implemented for holomorphic cusp forms for Γ_{0}(D). Our techniques include Voronoi summation, weighted exponential sums, and asymptotics expansions of Bessel functions. We then use these estimates in a new application of resonance sums. In particular we show that given only limited information about a Maass cusp form f (in particular a ﬁnite list of high Fourier coeﬃcients), one can determine its level and estimate its spectral parameter, and thus its Laplace eigenvalue. This is done using a large parallel computing cluster running MATLAB and Mathematica

#### Public Abstract

Prime numbers are numbers greater than 1 that are divisible only by 1 and themselves, such as 2, 3, 5, 7 and 11. Because any whole number can be written uniquely as the product of prime numbers, prime numbers are the fundamental building blocks in mathematics. Therefore it is of great importance to understand the prime numbers. In particular, how are the prime numbers distributed? Are they separated uniformly, or at random?

The goal of number theory is to answer questions like these. In this paper we investigate a function which exhibits many of the same properties as the prime numbers. We study when such functions can occur, and what properties they must have. Finally, using computational techniques and given only a small amount of information, we are able to determine key properties of these functions.

#### Keywords

publicabstract, automorphic forms, laplace eigenvalue, maass forms, number theory, resonance

#### Pages

viii, 60

#### Bibliography

58-60

#### Copyright

Copyright 2016 Paul Savala