  In this Ph.D dissertation (University of Virginia, 2022), we prove results
about the coefficients of partition-theoretic generating functions and of
coefficients of integer weight modular forms. Using various forms of the circle
method, we prove results about the distribution of partition statistics in
residue classes modulo $t$. For example, we prove that the parts of partitions
into distinct parts are equidistributed modulo $t$ (but that certain biases
occur nonetheless) and that the number of $t$-hooks in a partition is generally
not equidistributed modulo primes. We also obtain exact formulas for the
$t$-hook counting functions using modular transformation laws. We also employ
the circle method to prove a conjecture of Coll, Mayers and Mayers that the
$q$-series $(q, -q^3; q^4)_\infty^{-1}$ has non-negative coefficients. These
topics cover Chapters 3-6. Chapter 7 gives an application of partition
asymptotics for proving hyperbolicity of Jensen polynomials using the criterion
of Griffin, Ono, Rolen and Zagier. Chapter 8 gives a new method for solving
equations of the form $a_f(n) = \alpha$, where $\alpha \in \mathbb{Z}$ is odd
and $a_f(n)$ are the coefficients of a normalized Hecke eigenform with trivial
mod 2 Galois representation. The method is based on the primitive prime divisor
theorem of Bilu, Hanrot, and Voutier along with methods in effective algebraic
geometry for elliptic curves, hyperelliptic curves, and Thue equations.
