Non-existence of Ramanujan congruences in modular forms of level four

Ramanujan famously found congruences for the partition function like p(5n+4) = 0 modulo 5. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Gamma_{1}(4) which is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.Comment: 19 page

Efficient computation of the overpartition function and applications

In this paper we develop a method to calculate the overpartition function efficiently using a Hardy-Rademacher-Ramanujan type formula, and we use this method to find many new Ramanujan-style congruences whose existence is predicted by Treneer and a few of which were first discovered by Ryan, Scherr, Sirolli and Treneer

Asymptotics, Equidistribution and Inequalities for Partition Functions

This thesis consists of three research projects on asymptotics, equidistribution properties and inequalities for partition and overpartition functions. We start by proving that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for $n$ large enough, the two quantities are different, and that which of the two is bigger depends on the parity of $n.$ By doing so, we answer a conjecture formulated by Bringmann and Mahlburg (2012). We continue by placing this problem in a broader context and by proving that the same results are true for partitions into any powers. For this, we invoke an estimate on Gauss sums found by Banks and Shparlinski (2015) using the effective lower bounds on center density from the sphere packing problem established by Cohn and Elkies (2003). Finally, we compute asymptotics for the coefficients of an infinite class of overpartition rank generating functions, and we show that $\overline{N}(a,c,n),$ the number of overpartitions of $n$ with rank congruent to $a$ modulo $c,$ is equidistributed with respect to $0\le a< c,$ as $n\to\infty,$ for any $c\ge2.$ In addition, we prove some inequalities between ranks of overpartitions recently conjectured by Ji, Zhang and Zhao (2018), and Wei and Zhang (2018)

Mock theta functions and asymptotics for partition-theoretic functions

This thesis contains research articles on various topics in the theory of mod- ular forms and integer partitions. First we revisit Ramanujan’s original defini- tion of a mock theta function. We solve the general problem of understanding Ramanujan’s definition explicitly for the universal mock theta function g3, an- swering a question of Rhoades. After that we study a new spt function and its crank function. We investigate asymptotic aspects of this crank function and confirm a positivity conjecture of the crank. We further analyze a sign pattern of the crank and obtain linear congruences of the spt function via its mock modularity. Finally we provide the asymptotic formula for so-called odd-even partitions whose generating function appears in Ramanujan’s identities. We also study their overpartition analogue odd-even overpartitions

Mock modular forms as pp-adic modular forms

In this paper, we consider the question of correcting mock modular forms in order to obtain $p$-adic modular forms. In certain cases we show that a mock modular form $M^+$ is a $p$-adic modular form. Furthermore, we prove that otherwise the unique correction of $M^+$ is intimately related to the shadow of $M^+$.Comment: 17 Page