#### Document Type

Dissertation

#### Date of Degree

Spring 2016

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Yangbo Ye

#### Abstract

Consider either (i) *f *= *f*_{1 }⊠ *f*_{2 }for two Maass cusp forms for SL_{m}(ℤ) and SL_{m′}(ℤ), respectively, with 2 ≤ m ≤ m′, or (ii) *f*= *f*_{1 }⊠ *f*_{2 }⊠ *f*_{3 }for three weight 2*k* holomorphic cusp forms for SL_{2}(ℤ). Let λ_{f}(n) be the normalized coefficients of the associated L-function L(s, *f*), which is either (i) the Rankin-Selberg L-function L(s, *f*1 ×*f*2), or (ii) the Rankin triple product L-function L(s, *f*1 ×*f*2 ×*f*3). First, we derive a Voronoi-type summation formula for λ_{f} (n) involving the Meijer G-function. As an application we obtain the asymptotics for the smoothly weighted average of λ_{f} (n) against *e*(αn^{β}), i.e. the asymptotics for the associated resonance sums. Let ℓ be the degree of L(s, *f*). When β = 1/ℓ and α is close or equal to ±ℓq 1/ℓ for a positive integer *q*, the average has a main term of size |λ_{f} (*q*)|X ^{1/2ℓ+1/2} . Otherwise, when α is fixed and 0 < β < 1/ℓ it is shown that this average decays rapidly. Similar results have been established for individual SL_{m}(ℤ) automorphic cusp forms and are due to the oscillatory nature of the coefficients λ_{f} (n).

#### Public Abstract

Automorphic forms can be thought of as a generalization of classical trigonometric and elliptic functions. The latter are periodic functions defined on the complex numbers, whereas the former are invariant functions defined on more general topological groups. Examples of automorphic forms include modular forms and Maass forms, both of which are discussed in this thesis. These automorphic forms all admit a Fourier expansion because they are well-behaved and periodic. Moreover, attached to each one of these forms is a special function, called the *L*-function, which is defined as a series involving the Fourier coefficients. These *L*-functions are generalizations of important fundamental number theoretic objects like the Riemann zeta function and the Dirichlet *L*-series.

It is possible to combine automorphic forms in a variety of ways to construct new forms. One of the most straightforward constructions is to take the product of two or more automorphic forms. Unfortunately, it is not known whether or not the resulting form is automorphic, although this is suspected to be true and is known in a few select cases. This problem is referred to as the Langlands Functoriality Conjecture and remains one of the most difficult unsolved problems in modern number theory. This thesis establishes a collection of summation formulas involving the *L*-series coefficients attached to the product of several automorphic forms. As an application it is shown that these *L*-series coefficients have the exact properties that one would expect if the products were indeed automorphic.

#### Keywords

publicabstract, exponential sums, Fourier-Whittaker, Meijer G-function, Rankin-Selber, resonance sums

#### Pages

vi, 55

#### Bibliography

53-55

#### Copyright

Copyright 2016 Kyle Czarnecki