The Recursive 5130 Arithmetic Circuit and Robin's Inequality: Exact Extremal Prime-power Dynamics,large-scale Numerical Margins, and Critical-line Criteria
Year: 2026
Type: article
Abstract: We study Robin’s inequality primarily through explicit colossally abundant extremizers, with the recursive reduced-residue circuit modulo $5130 = 2 \cdot 3^3 \cdot 5 \cdot 19$ as a systematic accounting of prime events and their complementary composite layers. We prove an exact arithmetic decomposition, give a complete local exponent-threshold construction of colossally abundant maximizers for each positive extremal parameter, and derive a logarithmic interpolation principle for entire intervals between consecutive extremizers. Reproducible high-precision computations cover selected extremizers up to 72,284 decimal digits, with 15,201 distinct prime factors and a positive Robin deficit of approximately 0.0034990899494 in the largest case. A plotted sequence shows that the margins are positive in the sampled cases but diminish substantially at large scales. We give the precise implication from a universal endpoint inequality to the Riemann Hypothesis, and provide brief complementary Li–Weil and spectral observations.
Author: Paltoo, Nigel.S.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225780075
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