Date of Award


Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Fabrizio Zanello

Committee Member 1

William J. Keith

Committee Member 2

Melissa S. Keranen

Committee Member 3

John A. Jaszczak


The purpose of this dissertation is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities is that, under suitable existence conditions, for any $t$ coprime to $3$, if the $t$-multipartition function is odd with positive density, then $p(n)$ is also odd with positive density. Additionally if \emph{any} $t$-multipartition function is odd with positive density, then either $p(n)$ or the $3$-multipartition function (or both) are odd with positive density. All of these facts appear virtually impossible to show unconditionally today. Our arguments employ a combination of algebraic and analytic methods, including certain technical tools recently developed by Radu in his study of the parity of the Fourier coefficients of modular forms.

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