Party Results For Partition Function Sums And Congrunences For Appell-Lerch Sums

The parity on partition function is a difficult problem in the theory of integer partitions and many well-known combinatorialists are interested in it.Appell-Lerch sums were introduced by Appell and Lerch and this function plays important role in q-series since it is closely related to many mathematical topics such as Ramanujan's mock theta functions and the ranks of integer partitions.In this thesis,we study the parity for sums of partition functions and congruences for Appell-Lerch sums.Recently,Ballantine and Merca proved that if (?) if and only if bn(10)1 is a square.In this thesis,we investigate septuple (?) for which(?) if and only if (?)is a square.We prove some new parity result of sums of partitions and squares in arithmetic progressions which are analogous to the result due to ballantine and merca.In 2012,Chan proved some congruences and gave several conjectures for Appell-Lerch sums.Recently,Qu,Wang and Yao proved congruences modulo 2which generalize some conjectures given by Chan.In this thesis,we prove new congruences modulo 4 and 5 for Appell-Lerch sums.