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arXiv · 2510.10093

Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values

Sumukha Sathyanarayana·N. Guru Sharan
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Abstract

In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $\Theta(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations of $\Theta(r,s,t,x)$ 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.

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Sumukha Sathyanarayana, N. Guru Sharan. 2025-10-11. Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values. https://arxiv.org/abs/2510.10093

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