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

Kam Cheong Au

TL;DR

The paper develops a unified Mellin-transform kernel method to analyze Witten zeta functions for rank-2 and rank-3 root systems, enabling meromorphic continuation, pole-residue structure, and explicit special values. It introduces an explicit integral kernel framework, derives a rank-2 and a rank-3 convolution formalism, and applies creative telescoping to obtain closed-form integrals that determine derivatives at the origin and positive-pole residues. The main results include precise pole locations for $\xi_f(s)$, closed-form residues (often gamma-valued) at abscissas of convergence, and explicit expressions for $\xi_f(-n)$ and $\xi_f(0),\xi_f'(0)$ for rank-2 and rank-3 cases, as well as asymptotic formulas for the number of representations $r_\Phi(n)$. These findings reveal deep connections to Eisenstein series and $p$-adic phenomena, and they yield sharp asymptotics for representation counts, with potential generalization to higher rank root systems.

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.

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

TL;DR

The paper develops a unified Mellin-transform kernel method to analyze Witten zeta functions for rank-2 and rank-3 root systems, enabling meromorphic continuation, pole-residue structure, and explicit special values. It introduces an explicit integral kernel framework, derives a rank-2 and a rank-3 convolution formalism, and applies creative telescoping to obtain closed-form integrals that determine derivatives at the origin and positive-pole residues. The main results include precise pole locations for , closed-form residues (often gamma-valued) at abscissas of convergence, and explicit expressions for and for rank-2 and rank-3 cases, as well as asymptotic formulas for the number of representations . These findings reveal deep connections to Eisenstein series and -adic phenomena, and they yield sharp asymptotics for representation counts, with potential generalization to higher rank root systems.

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 -adic observation.

Paper Structure

This paper contains 26 sections, 41 theorems, 365 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Derivative at origin
  3. Poles of $\xi_f(s)$
  4. Values of $\xi_f(-n), n\in \mathbb{N}$
  5. Residue of $\xi_\Phi(s)$ at abscissa of convergence
  6. Integration kernel (rank 2)
  7. Meromorphic continuation of $\xi_f(s)$
  8. $\xi_f(s)$ is moderately increasing
  9. Values of $\xi_f(s)$ at non-positive integers
  10. A connection to Eisenstein series
  11. $p$-adic properties of $\zeta_{B_2}(-2n-1)$ and $\zeta_{G_2}(-2n-1)$
  12. Poles of $\xi_f(s)$
  13. A class of convolution integral and calculation of $\xi_f'(0)$
  14. $\mathcal{A}(k,s,\alpha)$ in terms of Hurwitz zeta functions.
  15. A class of convolution integral
  16. ...and 11 more sections

Key Result

Theorem 1.4

For rank $2$ and $3$ root systems, $r_\Phi(n)$ has the following form here $C$ and $a_j$ are explicit constants that depend on $\Phi$ (see Section 11 for their explicit values). Alsothe astute readers might wonder why $0$ is included in some $J$, since it could be absorbed into the constant $C$, the reason for this will become clear later.,

Figures (7)

  • Figure 1: The contour $C(1)$.
  • Figure 2: The contour $C(\infty)$.
  • Figure 3: The singularity diagram when $d=2$. Its shape remains essentially the same for other $d$. The two red lines intersect at $\operatorname{Re}(s) = 2/(d+2)$.
  • Figure 4: Singularity diagram of $\Gamma(z)\zeta(z) \alpha^{-z}\Gamma(s-z) \zeta(s-z)$.
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (108)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • ...and 98 more