The Mordell-Tornheim zeta function: Kronecker limit type formula, Series Evaluations and Applications

TL;DR

This paper develops Kronecker limit formulas for the Mordell-Tornheim zeta function, connecting it to classical modular relations and deriving new series evaluations and functional equations.

Contribution

It introduces Kronecker limit type formulas for the Mordell-Tornheim zeta function and unifies various modular relations under a new perspective.

Findings

Established Kronecker limit formulas for the zeta function.

Derived series evaluations involving classical functions.

Discovered a new family of mixed functional equations.

Abstract

In this paper, we establish Kronecker limit type formulas for the Mordell-Tornheim zeta function $\Theta(r,s,t,x)$ 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 $\Theta(r,s,t,x)$ 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.