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

TL;DR

This work links Kronecker limit formulas for the generalized Mordell-Tornheim zeta function $\\Theta(r,s,t,x)$ to two-term functional equations of the Herglotz--Zagier family $F_r(x)$, establishing equivalence criteria that unify key modular relations. A Kronecker limit type expansion for $\\Theta(r,r,t,x)$ around $t=1-r$ is derived, enabling the demonstration that the two-term equations for $F(x)$ and $F_r(x)$ are equivalent to the respective limit formulas for $\\Theta(1,1,t,x)$ and $\\Theta(r,r,t,x)$. The authors further develop the MT-zeta perspective to reproduce Guinand’s and Ramanujan’s modular relations and to present a cohesive table tying various classical results under one umbrella. These results illuminate how Mordell-Tornheim zeta functions centralize modular relations in the literature and offer a path toward higher-order generalizations and three-term analogues with potential broader applicability in analytic number theory. The paper thus provides a unified methodology for deriving and interpreting modular-type relations via Kronecker limit phenomena in generalized zeta settings.

Abstract

We establish Kronecker limit type formula for the generalized Mordell-Tornheim zeta function $Θ(r,r,t,x)$ 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 $Θ(1,1,t,x)$ 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.

Figures (1)

• Figure 1: The table shows the connections between the Mordell-Tornheim zeta function and the various modular relations in literature.

Theorems & Definitions (20)

• Theorem 1.1
• remark 1
• Theorem 1.2
• remark 2
• proof : Theorem \ref{['KLF Theorem']}
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Proposition 3.3