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 \[ \varphi_{\delta,\boldsymbol\rho}(s) = \frac12+\delta + \delta\sum_{j=1}^d\rho_jp_j^{-s}, \qquad \boldsymbol\rho\in B_d, \] the renormalized positive coefficient operators converge uniformly in operator norm, with \(O(\delta)\) error, to an explicit multivariate weighted Hankel operator \(\mathcal H_{\boldsymbol\rho}\); consequently, \[ 2\delta\|C_{\varphi_{\delta,\boldsymbol\rho}}\|^2 = \|\mathcal H_{\boldsymbol\rho}\| + O(\delta) \] uniformly over \(B_d\). We show that the limiting operator admits the total-degree reduction \[ \mathcal H_{\boldsymbol\rho} \simeq D_{\boldsymbol\rho} H_{R_{\boldsymbol\rho}/2} D_{\boldsymbol\rho}\oplus\mathbf{0}, \] 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 \(\|\mathcal H_{\boldsymbol\rho}\|\), 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 \(\sigma>\frac12\), 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-\(\sigma\) perturbative regime, and certified finite-dimensional approximation.