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

2024/12/23 by Kam Cheong Au, Au, Kam Cheong · 1 citation
Mathematics · #11M32 #11M35 (Primary) 11P82 #33C220 (Secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2412.17196

openalex publication_date 2024/12/23 · openalex created_date 2024/12/25 · openalex updated_date 2026/07/28

Abstract

By introducing a novel integration kernel for 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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• Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values

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