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

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 $${\phi }_{\delta ,\mathit{\rho }}(s)=\frac{1}{2}+\delta +\delta \sum _{j=1}^d{\rho }_j{p}_j^{-s},\mathit{\rho }\in B_d,$$ the renormalized positive coefficient operators converge uniformly in operator norm, with $O(\delta )$ error, to an explicit multivariate weighted Hankel operator ${\mathcal{H}}_{\mathit{\rho }}$; consequently, $$2\delta \mathrm{\parallel }{C}_{{\phi }_{\delta ,\mathit{\rho }}}{\mathrm{\parallel }}^2=\mathrm{\parallel }{\mathcal{H}}_{\mathit{\rho }}\mathrm{\parallel }+O(\delta )$$ uniformly over $B_d$. We show that the limiting operator admits the total-degree reduction $${\mathcal{H}}_{\mathit{\rho }}\simeq D_{\mathit{\rho }}{H}_{R_{\mathit{\rho }}\mathrm{/}2}{D}_{\mathit{\rho }}\oplus \mathbf{0},$$ 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 $\mathrm{\parallel }{\mathcal{H}}_{\mathit{\rho }}\mathrm{\parallel }$, 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 $\sigma >\frac{1}{2}$, 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.