Equivalence criteria for the two--term functional equations for Herglotz--Zagier functions

We establish Kronecker limit type formula for the generalized Mordell-Tornheim zeta function $\Theta(r,r,t,x)$ as a function of the third argument around $t=1-r$. We then show that the above Kronecker limit type formula is equivalent to the two-term functional equation for the higher Herglotz function obtained by Vlasenko and Zagier. We also show the equivalence between a previously known Kronecker limit type formula for $\Theta(1,1,t,x)$ around $t=0$ and the two-term functional equation for the Herglotz-Zagier function obtained by Zagier. Using the theory of the Mordell-Tornheim zeta function, we obtain results of Ramanujan, Guinand, Zagier, and Vlasenko-Zagier as consequences, to further show that the Mordell-Tornheim zeta function lies centrally between many modular relations in the literature, thus providing the means to view them under one umbrella.