  In this Ph.D. dissertation (2018, Emory University) we prove theorems at the
intersection of the additive and multiplicative branches of number theory,
bringing together ideas from partition theory, $q$-series, algebra, modular
forms and analytic number theory. We present a natural multiplicative theory of
integer partitions (which are usually considered in terms of addition), and
explore new classes of partition-theoretic zeta functions and Dirichlet series
-- as well as "Eulerian" $q$-hypergeometric series -- enjoying many interesting
relations. We find a number of theorems of classical number theory and analysis
arise as particular cases of extremely general combinatorial structure laws.
Among our applications, we prove explicit formulas for the coefficients of the
$q$-bracket of Bloch-Okounkov, a partition-theoretic operator from statistical
physics related to quasi-modular forms; we prove partition formulas for
arithmetic densities of certain subsets of the integers, giving $q$-series
formulas to evaluate the Riemann zeta function; we study $q$-hypergeometric
series related to quantum modular forms and the "strange" function of
Kontsevich; and we show how Ramanujan's odd-order mock theta functions (and,
more generally, the universal mock theta function $g_3$ of Gordon-McIntosh)
arise from the reciprocal of the Jacobi triple product via the $q$-bracket
operator, connecting also to unimodal sequences in combinatorics and quantum
modular-like phenomena.
