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 \[ φ_{δ,\boldsymbolρ}(s) = \frac12+δ+ δ\sum_{j=1}^dρ_jp_j^{-s}, \qquad \boldsymbolρ\in B_d, \] 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, \[ 2δ\|C_{φ_{δ,\boldsymbolρ}}\|^2 = \|\mathcal H_{\boldsymbolρ}\| + O(δ) \] 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.