Research Article · Preprint Mathematics
Optimal energy for Dirichlet repairs in divisor bands
- Primary subject
- Mathematics
- Subject subfields
- Number Theory · Functional Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
We construct finite arithmetic corrections that impose an exact complex Dirichlet value using a prescribed band of divisor coordinates. For q = 1/2 − a + iT, 0 < a < 1/2, and the critical Bernoulli penalty ω(r) = r log²(er), the minimum energy for value t on N < r ≤ 2N is of order |t|² N^(3−2a) log²(eN), with explicit constants and onset independent of T. The minimizing correction has complete native norm O_a(|t| N^(−a)) and first absolute coefficient moment O_a(|t| N^(3/2−a)). Applied to native approximants at a smaller scale, it gives actual finite anchored and q-null trials with energy O_a(N^(3−a) log²(eN)) and an explicit approximation bound relative to the best native trial at the smaller scale. At a specified height-dependent cutoff, their complete compensated mean, including its boundary term, is O_a(N^(−3/2−5a/2)). The approximation limit is the optimal native distance, uniformly in T.
Keywords Dirichlet polynomials · divisor coordinates · minimum energy · Nyman–Beurling approximation · Bernoulli covariance
Scope of the work
Exact minimum energy within a prescribed divisor band, height-uniform two-sided capacity bounds, and simultaneous complete native-norm, coefficient-moment and compensated-mean estimates for the same finite correction. The approximation family converges uniformly in height to the optimal native distance; the cited classical decay rate is stated under RH.
Reproduction notes
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AI assistance supported derivation, literature comparison, independent source review and computational checks. The author is responsible for the mathematical statements and presentation.
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Cite this edition
Scott Schubr. (2026). Optimal energy for Dirichlet repairs in divisor bands (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/optimal-energy-dirichlet-repairs-divisor-bands/v/1
BibTeX
@techreport{luminary_optimal_energy_dirichlet_repairs_divisor_bands_v1,
title = {Optimal energy for Dirichlet repairs in divisor bands},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/optimal-energy-dirichlet-repairs-divisor-bands/v/1}
}Loading discussion…