AI 中文总结
本文研究膨胀欧拉乘积商的单位根截面，确定极点位置与留数，揭示奇偶性定律、整除性及CM逼近界，并证明振荡误差指数尺度精确。
AI 中文摘要
我们研究了由膨胀欧拉乘积构成的商的单位根截面。在边界点 \\(q=1\\) 处，奇数截面趋于非零的分圆常数，而偶数截面则具有无穷多个简单实极点。我们确定了所有足够靠后的极点的位置、间距和留数。在三次情形中，我们找到了一个初等近似中的主要振荡误差，并证明了其指数尺度是精确的。在原点处，我们确定了精确的接触阶，包括一个依赖于间隙的奇偶性定律，以及对于每个模 \\(|k-1|\ge2\\)，首项非零系数被确定的整数整除性。对于欧拉幂 \\(k\ge2\\)，一个移动的形式极点控制着小的固定模数下的截面误差；一个实的无零点区间给出了进一步的收敛区域。对于第二十四次幂，我们证明了显式的CM逼近界以及随着膨胀增加时的带符号渐近性。
英文摘要
We study root-of-unity sections of quotients formed from dilated Euler products. At the boundary point \(q=1\), odd sections tend to nonzero cyclotomic constants, whereas even sections have infinitely many simple real poles. We determine the locations, spacing, and residues of all sufficiently late poles. In the cubic case we find the leading oscillatory error in an elementary approximation and prove that its exponential scale is sharp. At the origin we determine exact contact, including a gap-dependent parity law, and integral divisibility with the first nonzero coefficient determined for every modulus \(|k-1|\ge2\). For Euler powers \(k\ge2\), a moving formal pole controls the section error at small fixed nomes; a real zero-free interval gives a further convergence regime. For the twenty-fourth power we prove explicit CM approximation bounds and a signed asymptotic as the dilation increases.
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