Optimal energy for Dirichlet repairs in divisor bands
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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.
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Available editions
- Edition 1 · 2026-10-26 · public
Research details
- Subjects
- Mathematics · Number Theory · Functional Analysis
- Keywords
- Dirichlet polynomials, divisor coordinates, minimum energy, Nyman–Beurling approximation, Bernoulli covariance
- Manuscript date
- 2026-10-26
- Reader reuse
- No additional permission granted
Scope
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.
Disclosures
AI assistance supported derivation, literature comparison, independent source review and computational checks. The author is responsible for the mathematical statements and presentation.
Reproduction
The publication package contains the complete manuscript and standalone LaTeX source, the 256-bit Arb finite verifier and reference output, the pinned python-flint dependency, reader instructions, citation metadata and a SHA-256 member manifest. See README.md for reproduction commands.