Shifted poles and chamber cancellation for classical Witten zeta functions
Abstract
We determine two infinite families of poles on the positive real axis for classical single-variable Witten zeta functions. In type Ar, for r ≥ 5, the point qrA = 2(r−4)/(r2+r−4) is a simple pole except in ranks $12$ and $20$, where it is a double pole. In type Dr, for r ≥ 4, the point qrD = (r−3)/(r(r−1)−1) is a simple pole except at D8, where it is double. In root-product normalization, we express the simple residues and the leading coefficients of these three double poles in terms of gamma, trigonometric, and Riemann zeta values. For r ≥ 4, the functions of types Br and Cr are holomorphic at qrBC = (r−3)/(r2−1) except possibly in ranks $7$ and $11$, where any pole is simple. These pole and holomorphy statements arise from quadratic normal Taylor coefficients. For every fixed higher even normal degree, we also determine exactly when the associated continued B/C chamber sum is nonzero; this auxiliary result does not by itself classify poles of the full Witten function. The proofs combine exhaustive support classifications, explicit integration-by-parts identities, and finite chamber relations. No numerical nonvanishing estimate is used.
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BibTeX
@misc{matuzas2026shifted,
title = {{Shifted poles and chamber cancellation for classical Witten zeta functions}},
author = {Jonas Matuzas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00290},
eprint = {2609.00290},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00290},
}
RIS
TY - UNPB
TI - Shifted poles and chamber cancellation for classical Witten zeta functions
AU - Jonas Matuzas
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00290
UR - https://arxiv.org/abs/2609.00290
AB - We determine two infinite families of poles on the positive real axis for classical single-variable Witten zeta functions. In type A_r, for r ≥ 5, the point q_r^A = 2(r−4)/(r^2+r−4) is a simple pole except in ranks $12$ and $20$, where it is a double pole. In type D_r, for r ≥ 4, the point q_r^D = (r−3)/(r(r−1)−1) is a simple pole except at D_8, where it is double. In root-product normalization, we express the simple residues and the leading coefficients of these three double poles in terms of gamma, trigonometric, and Riemann zeta values. For r ≥ 4, the functions of types B_r and C_r are holomorphic at q_r^BC = (r−3)/(r^2−1) except possibly in ranks $7$ and $11$, where any pole is simple. These pole and holomorphy statements arise from quadratic normal Taylor coefficients. For every fixed higher even normal degree, we also determine exactly when the associated continued B/C chamber sum is nonzero; this auxiliary result does not by itself classify poles of the full Witten function. The proofs combine exhaustive support classifications, explicit integration-by-parts identities, and finite chamber relations. No numerical nonvanishing estimate is used.
ER -