经典威滕ζ函数的移位极点与腔抵消
Shifted poles and chamber cancellation for classical Witten zeta functions
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中文总结 AI 辅助
该研究确定经典单变量威滕ζ函数正实轴上的两类无限极点族，推导相关留数与首项系数表达式，结合多种方法完成证明，未用数值非零估计。
中文摘要 AI 辅助
我们确定了经典单变量威滕ζ函数在正实轴上的两个无限极点族。在A型（$A_r$）中，当$r \geq 5$时，点$q_r^A = 2(r-4)/(r^2+r-4)$是单极点，仅在秩12和20时为二重极点；在D型（$D_r$）中，当$r \geq 4$时，点$q_r^D = (r-3)/(r(r-1)-1)$是单极点，仅在$D_8$时为二重极点。在根乘积归一化下，我们用伽马值、三角函数值和黎曼ζ值表示这些三个二重极点的单留数及首项系数。对于$r \geq 4$，B型（$B_r$）和C型（$C_r$）函数在$q_r^{BC} = (r-3)/(r^2-1)$处是全纯的，仅可能在秩7和11时出现单极点。这些极点与全纯性结论源于二次正则泰勒系数。对于每个固定的更高偶正则次数，我们还精确确定了相关的连续B/C腔和非零的条件；该辅助结果本身并不足以对完整威滕函数的极点进行分类。证明过程结合了穷举支撑分类、显式分部积分恒等式及有限腔关系，未使用任何数值非零估计。
英文摘要
We determine two infinite families of poles on the positive real axis for classical single-variable Witten zeta functions. In type $A_r$, for $r \geq 5$, 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 $r \geq 4$, 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 $r \geq 4$, 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.
