Mordell-Tornheim zeta function: Kronecker limit type formulas and special values

In this paper, we establish Kronecker limit type formulas for the generalized Mordell–Tornheim zeta function Θ(r, s, t, x) as a function of the third variable, in terms of Riemannzeta and Gamma values. We also give series evaluations of Θ(r, s, t, x) in terms of Herglotz–Zagier type functions, and their derivatives. As applications of this, we derive Kronecker limit type formula in the second variable and a new infinite family of modular relations called mixed functional equations. We also study the zeros, special values and singularities of the above function when all its arguments r, s and t are equal, which builds on a few earlier results due to Romik