The Mordell-Tornheim zeta function: Kronecker limit type formula, series evaluations and applications

In this paper, we establish Kronecker limit type formulas for the Mordell-Tornheim zeta function ?(r,s,t,x) as a function of the second as well as the third arguments. As an application of these formulas, we obtain results of Herglotz, Ramanujan, Guinand, Zagier and Vlasenko-Zagier as corollaries. We 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. We also give series evaluations of ?(r,s,t,x) in terms of Herglotz-Zagier function, Vlasenko-Zagier function and their derivatives. Using our new perspective of modular relations, we obtain a new infinite family of results called mixed functional equations.