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arXiv · 2501.01380

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

Sumukha Sathyanarayana·N. Guru Sharan
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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.

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Sumukha Sathyanarayana, N. Guru Sharan. 2025-01-02. The Mordell-Tornheim zeta function: Kronecker limit type formula, Series Evaluations and Applications. https://arxiv.org/abs/2501.01380

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