Executive Summary : | The project consists of two parts. First part contains problems related to transformation formulas associated to Wigert's type divisor function. Second part is about a problem related to partition theory namely plane over partition. PART A: Recently the PI along with her PhD student studied Wigert's type divisor function. Although the problem is a natural generalisation of the classical divisor problem, but there were very few studies on it. A generalisation of this Wigert type divisor function has been studied in connection with mock modular form. Recently with A. Dixit and R. Gupta, PI have found out that the Dirichlet series associated with it has a representation in terms of zeta function and infinite integral on the right half plane. So we intend to study 1) Does the Dirichlet series associated with it have meromorphic continuation throughout the complex plane? 2) What can be said about the transformation formulas associated with it (particularly in case of Lambert series associated with it) ? 3) we intend to study the Voronoi type summation formula associated with it. 4) Whether it can be studied further in connection with mock modular form (as studied by Mellit and Okada). PART B: In number theory and combinatorics the partition function of a positive integer n is the number of way of writing an integer as a sum of non-decreasing positive integers.In 1918, Hardy and Ramanujan initiated the analytic study of it with the use of the celebrated Hardy–Littlewood circle method. Since then the function has been studied extensively. This partition functions have been generalised by several mathematicians. S. Corteel et al. in 2011, gave generalization of partitions, called plane over partition, which have proven useful in several combinatorial studies of basic hypergeometric series. So the natural question comes to our mind: what about 1) The asymptotic of plane over partition and 2) We plan to study few other problems related to it. |