Number Theory Seminar
Speaker: Prof. Atul Dixit (IIT Gandhinagar)
Host: Krishnan Sivasubramanian
Title: Mordell-Tornheim zeta functions and functional equations of Herglotz-Zagier type functions
Time, day and date: 2:00:00 PM, Thursday, April 17
Venue: Ramanujan Hall
Abstract: In this talk, we will present our recent results on a generalization of the Mordell-Tornheim zeta function, in particular, the two- and three-term functional equations that it satisfies. This function is intimately connected with a new extension of the Herglotz-Zagier function F(x). The function F(x) is instrumental in Zagier's version of the Kronecker limit formula for real quadratic fields. Radchenko and Zagier recently studied arithmetic properties of F(x), in particular, their special values and functional equations coming from Hecke operators. One of our results on this extension not only gives the well-known two-term functional equation of F(x) as a special case but also those of Ishibashi functions, which were sought after for over twenty years. A grand generalization of an integral considered by Herglotz as well as its companion due to Muzzaffar and Williams, which involves generalized Fekete polynomials and character polylogarithms, is obtained. By deriving a functional equation for this generalization, we are able to get doubly infinite families of functional equations whose two special cases were recently obtained by Choie and Kumar. This is joint work with Sumukha Sathyanarayana and N. Guru Sharan.