| Ramanujan,a talented Indian mathematician,had made great contributions to the studies of integer partitions.The famous congruences and identities of p(n)obtained by Ramanujan have been a source of inspiration for many studies.Inspired by Ramanujan’s work,we study the arithmetic properties of odd ranks by dissecting their generating functions.With a similar argument,we also obtain the 2-dissection and 3-dissection of Ramanujan’s mock theta function ρ(q).Identities for the coefficients of ρ(q)follow from these dissections immediately.Classical results on p(n)will be given in Chapter 1.These are the research backgrounds of this thesis.We also introduce the recent studies on odd ranks and ρ(q).At the end of Chapter 1,we will give a brief introduction of the main results of the thesis.In Chapter 2,we established a 5-dissection formula for odd rank,we obtain the arithmetic properties of the odd ranks.In Chapter 3,we prove the 2-dissection and 3-dissection of ρ(q)from which identities on the coefficients of ρ(q)follow. |
