Critical Norm Profiles for Finite-Prime Composition Operators on the Hardy Space of Dirichlet Series

Abstract: We identify the critical boundary operator-norm profile of finite-prime composition operators on the Hardy--Hilbert space (\mathcal H^2) of Dirichlet series. For [ \varphi_{δ,\boldsymbolρ}(s) = \frac12+δ+ δ\sum_{j=1}^dρ_jp_j^{-s}, \qquad \boldsymbolρ\in B_d, ] the renormalized positive coefficient operators converge uniformly in operator norm, with (O(δ)) error, to an explicit multivariate weighted Hankel operator (\mathcal H_{\boldsymbolρ}); consequently, [ 2δ|C_{\varphi_{δ,\boldsymbolρ}}|^2 = |\mathcal H_{\boldsymbolρ}| + O(δ) ] uniformly over (B_d). We show that the limiting operator admits the total-degree reduction [ \mathcal H_{\boldsymbolρ} \simeq D_{\boldsymbolρ} H_{R_{\boldsymbolρ}/2} D_{\boldsymbolρ}\oplus\mathbf{0}, ] where the diagonal factors are convolution-collision norms of the normalized prime weights. This structure, together with the affine comparison principle of Brevig and Perfekt, yields an explicit concentration inequality for (|\mathcal H_{\boldsymbolρ}|), identifies the one-prime configurations as the exact equality cases in the limiting norm estimate, and gives a quantitative deficit away from them. For fixed (σ>\frac12), we also obtain a second-order expansion of the squared norm and fully finite-dimensional approximations with explicit total-degree and Dirichlet-sum truncation errors. Together, these results show that a single coefficient-operator structure governs the singular boundary profile, the fixed-(σ) perturbative regime, and certified finite-dimensional approximation.