Mordell-Tornheim zeta Functions and Functional Equations for Herglotz-Zagier type Functions

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), whose special values and functional equations coming from Hecke operators were recently obtained by Radchenko and Zagier. 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 unknown for about twenty years. Some interesting results on integrals involving logarithm and/or arctangent, and related to the generalized Mordell-Tornheim zeta function will also be given. This is joint work with Atul Dixit and N. Guru Sharan.