At the critical boundary of finite-prime composition operators, a rescaled squared norm approaches a direction-specific limiting profile, according to a proof-based analysis. The limit comes with an explicit concentration inequality: configurations that place the coefficient weight on a single prime coordinate are exact equality cases in the limiting norm estimate.
The work asks whether a singularly normalized critical family has a uniform operator-norm limit and how that limit reflects mass across prime coordinates. It studies finite-prime, zero-characteristic symbols with a fixed finite number d of prime variables. No data were used or generated; the subject is a family of mathematical operators rather than an empirical sample.
The profile at the boundary
To examine that question, the norm problem is reformulated through an equivalent coefficient operator and an explicit positive operator. The critical-limit proof uses the pole of the zeta function for the leading term, then controls the remainder with uniform Cauchy estimates and Hilbert–Schmidt bounds.
With δ tending to zero while the normalized direction ρ is held fixed, the result states that 2δ times the squared norm of the critical operator equals ∥Hρ∥ + O(δ), uniformly over B_d. In ordinary language, each normalized direction is associated with its own limiting operator Hρ; the error estimate is uniform across the stated coefficient region.
What coefficient concentration reveals
The direction-specific limit reflects how coefficient mass is distributed across prime coordinates. Its norm obeys an explicit concentration inequality, and one-prime configurations attain equality. For nonzero ρ, the deficit from λ(Rρ) is at least (1 − Q(ρ))(λ(Rρ) − 1), giving a quantitative lower bound when the mass is not fully concentrated.
To expose that structure, the analysis reduces Hρ by total degree. The resulting equivalence is Hρ ≃ Dρ H_{Rρ/2} Dρ ⊕ 0, a weighted Hankel model whose diagonal factors encode convolution-collision norms of the normalized prime weights. The calculation therefore follows how combinations of those weights collide under convolution while retaining their distribution in the diagonal factors.
In genuinely multivariate directions, the diagonal coefficients tend to zero at high degree and Hρ is compact. The one-prime case is different: distinct scaling paths toward the boundary can yield different limiting values. The selected path therefore matters for the one-prime limit.
A second asymptotic regime
The analysis also treats a small-coefficient regime with σ fixed above 1/2. As R tends to zero, the squared norm has the expansion ζ(2σ) + [ζ′(2σ)]²/ζ(2σ) · Σⱼ₌₁ᵈ rⱼ² + O(R⁴). The first correction is governed by the sum of squared coordinate coefficients, rather than by total coefficient mass alone.
For finite calculations, the analysis supplies a certified lower approximation to the squared norm. Under 2R < ε < 2σ − 1, the fully finite quantity λ_N,M(σ,r) satisfies 0 ≤ ∥Cφ∥² − λ_N,M(σ,r) ≤ E_N(σ,r,ε) + η_N,M(σ,r,ε). The error bound depends on the selected degree and Dirichlet-sum cutoffs.
A tightly bounded result
These results are confined to finite-prime, zero-characteristic symbols with a fixed finite d. The critical families use a fixed normalized direction, while the small-coefficient expansion is taken at fixed σ.
The document is an arXiv version-one preprint dated 26 Aug 2026. No data were used or generated, the authors declare no competing interests, and X. Fang was partially supported by NSTC, Taiwan, under grant No. 114-2115-M-A49-003-MY3.
Paper data and sources
Original title: Critical Norm Profiles for Finite-Prime Composition Operators on the Hardy Space of Dirichlet Series
Authors: Xiang Fang, Feng Guo, Aman Mishra, P. Muthukumar
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text