Arithmetic cutoff bounds and sharp tail capacity for local Nyman–Beurling certificates
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Abstract
We give complete arithmetic receivers for local Nyman–Beurling projection gain in the whole-line Poisson norm. Mixed-order Bernoulli covariances retain all complex phases and diagonalize in common divisor coordinates. A complete Farey-operator estimate with varying primitive amplitudes and the exact original weight gives uniform support N=o(T) on each fixed-gap pole-free wall. Combined with the compensated mean, it yields two-sided and additive enclosures for the actual optimized double-null gain using a corrected finite full model. For every fixed finite geometric module, optimization over arbitrary degrees and height-dependent coefficients still gives arbitrarily small true gain on positive-lower-density sets of actual heights; an explicit evaluation defect prices escape from these modules. Sharp fixed-trial gcd capacity, support-cost bounds and robust complete finite native examples remain complementary applications.
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Research details
- Subjects
- Mathematics · Number Theory · Complex Analysis · Functional Analysis
- Keywords
- Nyman–Beurling, Dirichlet polynomials, Bernoulli polynomials, mixed covariance, cotangent sums, gcd matrices, Poisson norms, Möbius inversion
- Manuscript date
- 2026-10-27
- Reader reuse
- No additional permission granted
Scope
Complete complex-q mixed-order arithmetic receivers; pole-free uniform N=o(T) cotangent/Farey return; two-sided and additive corrected optimized-gain comparison; degree-uniform fixed-geometric-module obstruction with positive-lower-density actual-height failure and paid evaluation escape; retained complete finite native applications.
Disclosures
AI assistants supported derivation, literature comparison, manuscript preparation and independent checks. The author is responsible for the mathematical statements and presentation.
Reproduction
The 34-page manuscript is supplied with complete editable sources and a 37-member publication package. Eleven portable verifiers reproduce the evaluation duals, exact null certificates, gcd capacity and tails, mixed-order receivers, compensated means, full-metric gain comparison, Farey estimates and geometric-module controls. All eleven were tested normally and under optimization from a clean extraction, with matching mathematical outputs. The manuscript gives the analytic infinite-tail and all-parameter proofs. Python 3.12.14 and python-flint 0.9.0 were tested.