## Dissertations, Master's Theses and Master's Reports

2018

#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Department of Mathematical Sciences

Fabrizio Zanello

William J. Keith

#### Committee Member 2

Melissa S. Keranen

John A. Jaszczak

#### Abstract

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.

Rights for JNT.pdf (65 kB)
Permissions for Material Used

COinS