Asymptotics for Traces of Weak Maass Forms and Applications
Abstract
In this dissertation, we give two main results. First, we use the Bruinier-Ono formula to give an asymptotic formula for the partition function $p(n)$ with an effective bound on the error term. Second, we give an asymptotic formula with a power saving error term for traces of a generic class of weight $0$ weak Maass forms. Those forms are images of weight $-2k$ weakly holomorphic modular forms of squarefree level $N$ under the differential operator $\mathcal{D}^k$. We apply this result to study the asymptotic distribution of several arithmetic functions, including the partition function $p(n)$, the Andrews' smallest parts function $\text{spt}(n)$, and the coefficients $\alpha(n)$ of Ramanujan's $f(q)$ mock theta function.
Citation
Khaochim, Narissara (2021). Asymptotics for Traces of Weak Maass Forms and Applications. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /193221.