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 \Theta(r,s,t,x) as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations of \Theta(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 zeroes, 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.