Master Dissertation Congruences For Partition Functions And Newman's Identities

As a major branch of number theory,the theory of integer partitions is a significant research field in combinatorics.It has a wide range of applications in group theory,probability theory,and mathematical physics.Congruence property of partition functions is one of the most active fields in the theory of partitions.In this thesis,by applying Newman's identities and some formulas for the representations of n as sums of k triangular numbers,we investigate congruence properties for two restricted partition functions:overpartition function and partitions with odd part distinct.We prove nonlinear congruences for overpartitions and make some progress on Mahlburg's conjecture.Overpartition is an important type of restricted partition function.It was first named by Lovejoy and Corteel.Later,congruences modulo powers of 2,3 and 5 have been discovered by Chen,Hirschhorn,Kim,Mahlburg,etc.In Chapter 2,we apply Newman's identities to establish congruences modulo 2048 of overpartition,while the current best result achieved was modulo1024.In addition,we establish some new nonlinear congruences modulo 2048.For Mahlburg's conjecture,we prove thatwhich(?)(n)denotes the number of overpartitions of n.For partitions with odd distinct part pod(n),we establish some relations between pod(n)and t_k(n),where t_k(n)is the number of representations of n as a sum of k triangular numbers.From the formula of t_k(n),new infinite families of congruences modulo 5 and 9 for pod(n)are proved,which generalize some results due to Radu and Sellers.