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]. The zeros of ζκ (s) are given by sn = exp(ρn/κ) where ρn are the zeros of ζ(s). Under the Riemann hypothesis, these zeros lie on the circle |s| = e1/(2κ). Scale prime numbers arise naturally as irreducible elements, with correspondence p = exp(ep/κ) to ordinary primes [8]. All results hold for any κ > 0 and are verified numerically with errors below 10−14. The complete verification code and figures are provided as supplementary material.