In this dissertation, we begin by presenting the result of F. K. C. Rankin and Swinnerton-Dyer on the location of the zeros of the Eisenstein series for the full modular group in the standard fundamental domain. Their simple but beautiful argument shows that all zeros are located on the lower...
We generalize overpartition rank and crank generating functions to obtain k-fold variants, and give a combinatorial interpretation for each. The k-fold crank generating function is interpreted by extending the first and second residual cranks to a natural infinite family. The k-fold rank generating functions generate two families of buffered Frobenius...
In this note, we generalize recent work of Mizuhara, Sellers, and Swisher that gives a method for establishing restricted plane partition congruences based on a bounded number of calculations. Using periodicity for partition functions, our extended technique could be a useful tool to prove congruences for certain types of combinatorial...
As indicated by the title The Collision of Quadratic Fields, Binary Quadratic Forms, and Modular Forms, this paper leads us to an understanding of the relationship between these three areas of study. In [4], Zagier gives the results of an intriguing example of the relationship between these areas. However there...
The first published notion that the j-function was in any way related to the Monster came in 1979, when Conway and Norton noted in [CN79] that each coefficient in the q- expansion of the j-function could be written as a (nontrivial) integral linear combination of the dimensions of irreducible representations...
Just as prime numbers can be thought of as the building blocks of the natural numbers, in a similar fashion, simple groups may be considered the building blocks of finite groups. Burnside considered the following
questions:
1. Do there exist non-abelian simple groups of odd order?
2. Do there exist...
In this paper we examine some of the developments concerning the Gauss class number problems and build a solid understanding of the class number. First we will develop some background knowledge necessary to understand the problem, specifically the theory of quadratic forms and quadratic fields and how the class number...
The study into specific properties of the partition function has been a rich topic for number theorists for many years. Much of the current work involving the arithmetic properties of the partition function and their seed in some keen observations of Ramanujan.In particular he discovered what are referred to as...
In 1956, Alder conjectured an integer partition inequality which generalized Euler’s partition identity, the first Rogers-Ramanujan identity, and a partition identity of Schur. Alder’s conjecture, proved in part by Andrews in 1971, followed by Yee in 2008, and finally completed by Alfes, Jameson, and Lemke Oliver in 2010 states that...
Following the work of Asai, Kaneko, and Ninomiya for Faber polynomials associated to the modular group, and Bannai, Kojima, and Miezaki's partial proof for the case of the Fricke group of level 2, we show that the zeros of certain modular functions for some low-level genus zero groups associated to...