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Kam Cheong Au

Publications and source records attributed to Kam Cheong Au.

13 recordsLinked to original sources
arXiv2026

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

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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arXiv2026

Combinatorics of hyperplane arrangements and Witten zeta function at the origin

We introduce a new method that brings the combinatorics of hyperplane arrangements into the study of representation zeta functions of compact Lie groups. For the Witten zeta function $ζ_Φ(s)$ associated with a root system $Φ$, our method yields elegant formulas for $ζ_Φ(0)$ and $ζ_Φ'(0)$ in terms of the exponents of various parabolic subsystems of $Φ$. Such formulas do not appear to be readily accessible through the conventional analytic techniques in the literature. More generally, the method applies to a broad family of conical zeta functions, expressing these two special values through the Möbius function of the intersection poset of the associated hyperplane arrangement.

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arXiv2026

Discovering hypergeometric series with harmonic numbers via Wilf-Zeilberger seeds

By extracting coefficients from Wilf-Zeilberger pairs with respect to auxiliary parameters, we discover many nontrivial hypergeometric series involving harmonic numbers. In particular, we obtain a rapidly convergent series for the depth-two multiple zeta value $ζ(5,3)$, which appears to be the first result of its kind in the literature. We also experiment with the Hilbert-Poincare series attached with a WZ-seed and conjecture that it admits a remarkably simple form, suggesting the presence of an underlying graded algebra structure behind WZ-seeds.

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arXiv2026

Iterated Integrals and Multiple Polylogarithm at Algebraic Arguments

By introducing a generalized notion of multiple zeta values associated with an arbitrary finite subset $S\subset \mathbb{P}^1(\mathbb{C})$ and studying their transformation properties under rational functions, we show that multiple polylogarithms evaluated at roots of unity (cyclotomic multiple zeta values, CMZVs) can be equivalently expressed in terms of iterated integrals involving certain non-roots of unity. We apply this theory to elucidate previously unknown $\mathbb{Q}$-linear relations among CMZVs: they come from nontrivial solutions of certain $S$-unit equations in the function field of $\mathbb{P}^1(\mathbb{C})$, thereby attaining the motivic dimension for low level and weight. We introduce a datamine of CMZVs that appears to be the first rigorous compilation of this kind in the literature. In addition, we formulate several nontrivial Galois descent conjectures for multiple polylogarithms and present applications to certain Apéry-type infinite series.

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arXiv2026

$q$-analogues of Wilf-Zeilberger seeds and Ramanujan $1/π^k$-formulas

We develop the notion of Wilf-Zeilberger seeds as a powerful framework for generating WZ-pairs and for lifting classical hypergeometric identities to the $q$-setting. As an application, we systematically obtain a large family of $q$-analogues of Ramanujan-type $1/π^k$ formulas, extending a body of literature previously limited to sporadic examples. While the majority of these $q$-analogues are modular, we also uncover several curious instances of mock modularity.

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arXiv2026

Vanishing of Witten zeta function at negative integers

We introduce a new analytic method for studying Witten zeta function of a root system $Φ$, based on a refined manipulation of an integral representation involving the Hurwitz zeta function. As an application, we prove high-order vanishing at negative even integers. This technique also describes non-trivially, the arithmetic nature of the leading term, in which the highest root of $Φ$ makes a surprising appearance.

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arXiv2024

Wilf-Zeilberger seeds and non-trivial hypergeometric identities

Through a systematic approach on generating Wilf-Zeilberger-pairs, we prove some hypergeometric identities conjectures due to Z.W. Sun, J. Guillera and Y. Zhao etc., including two Ramanujan-$1/π^4$, one $1/π^3$ formulas as well as a remarkable series for $ζ(5)$.

math.CO↗
arXiv2024

Creative telescoping and generating functions of (variants of) multiple zeta values

We show how to convert the generating series of interpolated multiple zeta values, or multiple $t$ values, with repeating blocks of length 1 into hypergeometric series. Then we invoke creative telescoping on their generating functions, in some known cases for illustration, and in some apparently new cases, reducing them to polynomials in Riemann zeta values. The new evaluations, including $ ζ^{1/2}(\{\bar2\}^n,3) $, $ ζ^\star(\{1,3\}^n,1,2) $ and $ t^{1/2}(2,\{1\}^n,2) $, resolve some questions raised elsewhere, and seem to be non-trivial using other methods.

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arXiv2023

Moments of ideal class counting functions

We consider the counting function of ideals in a given ideal class of a number field of degree $d$. This describes, at least conjecturally, the Fourier coefficients of an automorphic form on $\text{GL}(d)$, typically not a Hecke eigenform and not cuspidal. We compute its moments, and also investigate the moments of the corresponding cuspidal projection.

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