Research Projects

The projects have two parts. In Work Plan A, we will consider problems intimately connected to analytic number theory where we require some combinatorial tools to understand the problems. In Work Plan B, we will consider problems mainly related to combinatorial number theory where we need some analytic tools. Research Plan A: In this part of the proposal, we will study the asymptotic of the number of smallest parts in the power partition of a positive integer. As a continuation, we propose to study mean value estimates of the smallest parts in the power partition of a positive integer. Next project will be on a prime power partition in arithmetic progression. This will be a joint project with collaborator, Arindam Roy. These two projects will constitute our main research proposal. In a recent paper, PI and her PhD student studied Wigert's type divisor problem, or restricted divisor sums of a special kind. The method introduced in that paper opens up many related problems, and we believe a resolution to those problems can be made. We propose some problems in this direction. Research Plan B : For finite group G, Davenport constant D(G) was introduced (by K. Roger) as a motivation of prime ideal decompositions in algebraic number fields. A natural question is: What is its precise value for any finite abelian group? The answer is known for groups of rank 1, 2, and any p-groups. For a group with rank r greater than 2, its precise value is still unknown. Olson’s conjecture regarding its value, has been proven to be wrong for groups of rank 4,5 [Geroldinger and Scneider]. Our aim in this project is to classify the abelian groups and evaluate its value corresponding to each classification. One can also extend the definition of the Davenport constant for finite non-abelian groups, D’(G) and D"(G), which correspond to an unordered and ordered zero-sum subsequence, respectively. We want to study D’(G) for non-abelian groups, such as symmetric, Heisenberg groups. Dimitrov evaluated the value of D''(G) for the Heisenberg group. He conjectured that D"(G) = L(G) for any finite p-group, where L(G) is the Loewy length of group G. We would like to prove the above conjecture and study its behavior for other non-abelian groups. We are also interested in studying generalized Davenport constants. The Co-PI and her collaborator introduced the Davenport constant for random sequence. Such a study opens up a new horizon in the field of zero-sum theory, and we propose to study it further during this project. We also want to focus on the Harborth constant of finite groups. Its value is unknown even for abelian groups of rank 2. So we want to create an algorithm so that we can know the value of the Harborth constant for any group. Also, we would like to study this invariant theoretically for finite groups.