The Scale Group: A Novel Abelian Group Structure on the Positive Reals With Connections to Zeta Functions and Prime Numbers
We introduce an abelian group structure on the positive real numbers via the operation a ⊗κ b = exp(κ ln a ln b) for a parameter κ > 0. The transformation Tκ (x) = ln(κ ln x) establishes a group iso- morphism (M>1κ , ⊗κ ) ∼= (R, +), enabling harmonic analysis on the scale group. We define generalized zeta functions ζκ (s) = ∑ n−⊗κ s and prove ζκ (s) = ζ(κ ln s) [11 , 13]. […]