Arithmetic cutoff bounds and sharp tail capacity for local Nyman–Beurling certificates
Loading original first page…
Abstract
We determine the sharp tail capacity of anchored arithmetic trials in the whole-line Poisson norm at λ=α+iT, 1/2<α<1. After a prescribed exterior null is imposed, the complete norm has an M⁻¹ tail with a positive gcd-variance coefficient and an explicit period-free remainder. The normalized coefficient has exact infimum [12(ζ(2)ζ(2α)/|ζ(1+λ)|²−1)]⁻¹. Finite constrained capacities approach this limit at an explicit rate uniform in T. A dyadic factor imposes the null while preserving arbitrarily much projection gain at a paid support cost, yielding complete finite arithmetic lower certificates. Complementarily, an exact weighted divisor dual gives g_N(λ)≤K_α N^(2−2α)|ζ(λ)|² for all anchored complex coefficients supported on the integers at most N. The power is exact for the unrestricted head reconstruction norm. Montgomery's unconditional small-value theorem then gives a necessary support-cutoff growth rate for an all-height positive gain. For the same trials we also give a phase-preserving, finite-order expansion of the complete mean tail. Matching its two power kernels cancels the lower endpoint and improves the sharp fixed-trial mean order to M^(−α−1). Explicit corrected rows and full norm envelopes yield a constrained robust certificate with a unique scalar optimum. The summation and optimization machinery is classical; the arithmetic correction retains both nulls and the actual common head.
Read the full paper in the original PDF.
Edition files
- main.pdf150.0 KiB · pdf
SHA-256
43df4cb4a35e394154b108686059fc15547edc3e0051b2ee9bd49398be3b508c - arithmetic_cutoff_tail_capacity_publication.zip179.6 KiB · package
SHA-256
7104254763fe8e3edcacab8efe587f52b599ebc09390aaf3eada2442452178a4
Available editions
Research details
- Subjects
- Mathematics · Number Theory · Complex Analysis · Functional Analysis
- Keywords
- Nyman–Beurling, Dirichlet polynomials, Möbius inversion, gcd matrices, Poisson norms, Riemann zeta function
- Manuscript date
- 2026-10-26
- Reader reuse
- No additional permission granted
Scope
For anchored complex arithmetic trials at λ=α+iT with 1/2<α<1 and the specified exterior null, the paper proves exact gcd tail capacity, a uniform finite-cutoff rate, a weighted divisor support bound, phase-preserving mean-tail expansion and a uniquely optimized robust certificate.
Disclosures
AI assistants supported derivation, literature comparison, manuscript preparation and independent checks. The author is responsible for the mathematical statements and presentation.
Reproduction
The 16-page manuscript and main.tex are supplied in a 13-member package with README.md, CITATION.bib, MANIFEST.json, SHA256SUMS.txt, requirements.txt and six finite verifiers. README.md gives build and replay commands; python-flint 0.9.0 supports 256-bit Arb/Acb enclosures. The programs check weighted duals, null adaptation, gcd tails, finite capacities, compensated means and constrained certificates. Analytic tail and all-support claims are established in the manuscript; the finite examples retain complete-tail enclosures.