Root-of-Unity Sections of Dilated Euler-Product Quotients

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.