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arXiv · 2412.17196

On single-variable Witten zeta functions of rank two and three

Kam Cheong Au
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Abstract

By introducing a novel integration kernel for the Mellin transform, we uncover many previously unknown and intriguing properties of the Witten zeta functions of rank two and three. Detailed results concerning their pole locations, residues, and special values are obtained. We propose a non-trivial conjecture regarding their derivatives at the origin, which seems to encode deep information about the root system. We also discuss their behavior at negative integers, highlighting a connection with Eisenstein series and a $p$-adic observation.

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Kam Cheong Au. 2026-09-21. On single-variable Witten zeta functions of rank two and three. https://arxiv.org/abs/2412.17196

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