Abstracts for PLeaSANT (Participative Learning Seminar on Any Number Theory)

For modular forms, the Kaneko--Zagier equation is a second-order differential equation with respect to the Serre derivative. Analogously, for Jacobi forms, there exists an analogue of the Serre derivative that increases the weight of a Jacobi form by 2 and preserves its index. It is not difficult to describe the kernel of this operator, but finding solutions of even second-order differential equations, known as Kaneko--Zagier type equations, is not such a trivial problem. In my talk, I will present some current progress on this topic.

In 1976, Fujii obtained an asymptotic formula for an analogue of the Titchmarsh divisor problem for the product of two primes. Recently, Drappeau and Topacogullari studied an asymptotic formula for the Titchmarsh divisor problem with a small shift for almost primes. In this talk, we will discuss an asymptotic formula for the Titchmarsh divisor problem for the product of primes, which is uniform in the shift parameter.