Ranks Of Overpartitions Modulo 6 And 10

The theory of partitions is an important field which is related to com-binatorics,modular form and number theory.The statistics of partitions are main subjects in partition theory,especially“rank”.The rank was originally used to provided the combinatorial interpretations of Ramanujan's congru-ences.In this thesis,we will establish the generating functions for the ranks differences of overpartitions modulo 6 and 10,and obtain some inequali-ties between the ranks of overpartitions modulo 6 and 10.In light of these generating functions,some relations between mock theta functions and the ranks of overpartitions modulo 6 and 10 can also be found.Here's a brief introduction to the layout of the article:In Chapter 1,we give some background information about the rank differences problems and introduce some basic definitions used in this thesis.At the same time,some related classical theorems and definitions in modular forms and mock theta functions are introduced.In Chapter 2,we establish relationships between multiple generalized Lambert series and multiple infinite products.This links two different types of automorphic forms.Compared with Chan's work,these new identities are useful in generating various formulas for generalized Lambert series with similar poles.The objective of Chapter 3 and 4 is to derive the generating functions of D-ranks and M₂-ranks of overpartitions mod 6 and 10.For the generating functions mod 6,we need the generalized Lambert series identities obtained in Chapter 2 and Chan.Moreover,the theory of modular forms also be used for proved the identities of generating functions mod 10.In Chapter 5,we obtain some applications of the generating functions of D-rank differences and M₂-rank differences of overpartitions.First,we de-rive some inequalities from this generation functions.Then,the relations be-tween mock theta functions and this generating functions have been proved.