Number Theory

Three Greedy Representations of the MRB Constant

Authors: Marvin Ray Burns
Comments: 9 Pages.

We compare three constructive greedy representations of the MRB constant. Algorithm A usesreciprocal-kernel residue magnitudes, beginning with the unusually accurate approximation A(8).Algorithm B subtracts the original radical increments m1/m − 1 from the anchor B(16). AlgorithmC employs log(m)/m, seeded by the exact Dirichlet-eta value ηu2032(1) = γ log 2 − 12 (log 2)2. We proveconvergence for the specified selection rules and obtain a quadratic error bound for the nearest-residuemethod and a logarithmically improved quadratic bound for both one-sided methods. AlgorithmC also gives an exact infinite product for eCMRB . A common Lambert-W backbone generates theinteger indices: exact inverse formulas for B and C, and a rigorous, sub-unit-width Lambert bracketfor A. Five logarithmic corrections leave an error of approximately 6.85 × 10−105. These expansionsare not irrationality criteria; we do not claim invention of the general greedy principle.
Category: Number Theory