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

Sumukha Sathyanarayana, N. Guru Sharan

TL;DR

The paper develops Kronecker limit type formulas for the generalized Mordell-Tornheim zeta function $\Theta(r,s,t,x)$, first in the third variable $t$ and then in the second variable $s$, revealing detailed pole structures and principal parts that hinge on the parameters $(r,s,t)$ and the scale $x$. It unifies a spectrum of modular relations (Ramanujan, Guinand, Herglotz–Zagier, Vlasenko–Zagier) through a single framework and provides new mixed functional equations by expressing $\Theta$ via polylogarithms, Hurwitz zeta functions, and Herglotz–Zagier functions. The work yields novel series evaluations and a recursive machinery (Theorem molty) that recovers known two-term functional equations and generates infinite families of mixed identities, underscoring the central role of $\Theta$ as a unifying object in the modular-relations landscape. These results open avenues for three-variable expansions and higher-term functional equations with potential connections to Arakawa–Kaneko constants and related integrals. The methods blend partial fractions, inversion symmetries, and careful analysis of Laurent expansions to produce explicit expressions with applications to modular relations and zeta-function theory.

Abstract

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

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

TL;DR

The paper develops Kronecker limit type formulas for the generalized Mordell-Tornheim zeta function , first in the third variable and then in the second variable , revealing detailed pole structures and principal parts that hinge on the parameters and the scale . It unifies a spectrum of modular relations (Ramanujan, Guinand, Herglotz–Zagier, Vlasenko–Zagier) through a single framework and provides new mixed functional equations by expressing via polylogarithms, Hurwitz zeta functions, and Herglotz–Zagier functions. The work yields novel series evaluations and a recursive machinery (Theorem molty) that recovers known two-term functional equations and generates infinite families of mixed identities, underscoring the central role of as a unifying object in the modular-relations landscape. These results open avenues for three-variable expansions and higher-term functional equations with potential connections to Arakawa–Kaneko constants and related integrals. The methods blend partial fractions, inversion symmetries, and careful analysis of Laurent expansions to produce explicit expressions with applications to modular relations and zeta-function theory.

Abstract

In this paper, we establish Kronecker limit type formulas for the Mordell-Tornheim zeta function 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 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.
Paper Structure (13 sections, 15 theorems, 148 equations, 3 figures)

This paper contains 13 sections, 15 theorems, 148 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Modular relations in the literature
  3. Kronecker Limit type formula for $\Theta(r,s,t,x)$ in the Third variable
  4. Applications in modular relations
  5. Vlasenko and Zagier's functional equation
  6. Guinand's functional equations
  7. Ramanujan's functional equation
  8. Kronecker limit type formula for $\Theta(r,s,t,x)$ in the Second variable
  9. Mixed functional equations
  10. Kronecker limit type formula
  11. Modular relations: The Mordell-Tornheim zeta perspective
  12. Concluding Remarks
  13. Acknowledgment

Key Result

Proposition 2.1

Let $r,s,t \in \mathbb{C}$ such that $r \not\in \mathbb{N}$, and $\operatorname{Re}(t)> \textup{max} \, \{0,1- \operatorname{Re}(r), 1- \operatorname{Re}(s), 2-\operatorname{Re}(r)-\operatorname{Re}(s)\}$. For any $M \in \mathbb{N} \cup \{0\}$ and $x>0$, we have

Figures (3)

  • Figure 1: The cases for $r \in \mathbb{Z} \setminus \mathbb{N}$.
  • Figure 2: The cases for $r \in \mathbb{N}$.
  • Figure 3: The table shows the connections between the Mordell-Tornheim zeta function and the various modular relations in literature.

Theorems & Definitions (33)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 23 more