On the ranks of partitions modulo certain integers

Abstract

This thesis focuses on the rank of partition functions, identities related to generating functions of ranks modulo different integers and Ramanujan's convolution sum. Most results in Chapters 2 and 4 are reproduced from [14] and [10], respectively.

Ramanujan had three famous congruences for the partition function modulo 5, 7 and 11. F. J. Dyson defined the the rank of partitions and conjectured that ranks could provide combinatorial explanations for the cases of 5 and 7. A. O. L. Atkin and H. P. F. Swinnerton-Dyer proved his conjecture using generating functions for the rank difference modulo 5 and 7. From Theorem 8.16 in F. G. Garvan's paper [20], we know that results on dissections of the rank modulo m are equivalent to results on rank difference results modulo m, which inspired us to find a 3-dissection of ranks modulo 9 in Chapter 2. We also give an identity involving generating functions of ranks modulo 3 and 9.

Ramanujan recorded several entries which are related to generating functions of the rank modulo different integers. Finding analogous identities is the motivation of Chapter 3. We give some identities, some of which are obtained by using Ramanujan's entries.

In Ramanujan's paper [36], he proved a formula for convolutions of sum of divisors functions. In Chapter 4, we find formulas for convolutions of the sum of divisor functions twisted by the Dirichlet character, which are analogous to Ramanujan's.