Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values
Sumukha Sathyanarayana, N. Guru Sharan
TL;DR
This work develops Kronecker limit-type formulas for the generalized Mordell–Tornheim zeta function Θ(r,s,t,x) across its third and second variables, using Mellin–Barnes-type methods and partial-fraction/Herglotz–Zagier techniques, respectively. It provides explicit Laurent expansions near critical lines t = 1−r−ℓ and t = 2−r−s, uncovering pole structures that depend on the parameters r and s, and yields series evaluations in terms of zeta values, the digamma function, and its derivatives. A novel infinite family of mixed functional equations is derived, together with a second-variable Kronecker limit-type formula, enriching the connections to Guinand-type and Vlasenko–Zagier-type relations. The paper also analyzes special values and zeros of Θ(s,s,s,x), revealing a parity phenomenon in Θ(−j,−j,−j,x) and linking to Romik’s Witten-zeta results for SL(3). Overall, the results illuminate the analytic structure of Θ and suggest further links to root-system zeta values and modular-type relations.
Abstract
In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $Θ(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations of $Θ(r,s,t,x)$ in terms of Herglotz-Zagier type functions, and their derivatives. As applications of this, we derive Kronecker limit type formula in the second variable and a new infinite family of modular relations called mixed functional equations. We also study the zeroes, special values and singularities of the above function when all its arguments $r,s$ and $t$ are equal, which builds on a few earlier results due to Romik.