English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this Ph.D. dissertation (2018, Emory University) we prove theorems at theintersection of the additive and multiplicative branches of number theory,bringing together ideas from partition theory, $q$-series, algebra, modularforms and analytic number theory. We present a natural multiplicative theory ofinteger partitions (which are usually considered in terms of addition), andexplore new classes of partition-theoretic zeta functions and Dirichlet series-- as well as "Eulerian" $q$-hypergeometric series -- enjoying many interestingrelations. We find a number of theorems of classical number theory and analysisarise 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 statisticalphysics related to quasi-modular forms; we prove partition formulas forarithmetic densities of certain subsets of the integers, giving $q$-seriesformulas to evaluate the Riemann zeta function; we study $q$-hypergeometricseries related to quantum modular forms and the "strange" function ofKontsevich; 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 $q$-bracketoperator, connecting also to unimodal sequences in combinatorics and quantummodular-like phenomena.

Eulerian series, zeta functions and the arithmetic of partitions