Sharp regularity criteria for arithmetic cutoff functionals
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Abstract
We classify the coefficient penalties that control point evaluation and absolute coefficient moments of finite arithmetic cutoff certificates. In divisor coordinates, let W_ω = Σ_{r≥2} ω(r)(J₄(r) + 10J₂(r))|v_r|²/180, where ω(r) ≥ 1 is arbitrary. Every fixed integer Bernoulli cutoff value is bounded in this norm if and only if Σ_r ω(r)^(−1) < ∞. For every integer R ≥ 1, the absolute moment Σ_d d^R |c_d| is bounded if and only if Σ_r r^(2R−2)/ω(r) < ∞. Both classifications persist after any fixed finite collection of bounded homogeneous constraints. A full-divisor lower bound supplies necessity and settles the critical logarithmic exponents. Applications give an exact constrained point capacity, a norm-controlled compensated-mean remainder, and an explicit native approximation floor. At the critical moment penalties, the floor gives a quantitative logarithmic lower bound on the energy required for a given native error.
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Available editions
- Edition 1 · 2026-10-26 · public
Research details
- Subjects
- Mathematics · Number Theory · Functional Analysis
- Keywords
- Bernoulli polynomials, divisor inversion, weighted regularity, Nyman–Beurling approximation, sharp logarithmic endpoints
- Manuscript date
- 2026-10-26
- Reader reuse
- No additional permission granted
Scope
Necessary-and-sufficient summability criteria for integer Bernoulli cutoff evaluation and absolute coefficient moments under arbitrary divisor-coordinate penalties, with finite bounded-null stability and sharp critical logarithmic thresholds. Applications include constrained point capacities, compensated-mean remainder estimates and necessary native approximation energy bounds.
Disclosures
AI assistance supported derivation, literature comparison, manuscript preparation and computational checks. The author is responsible for the mathematical statements and presentation.
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