Riemann Heights and Decomposed Primes

This paper examines a transformation of the coordinates associated with prime numbers, defined by p₂₀ = p/c with c = 1 − 1/2²⁰ = 1 048 575/1 048 576, and its consequences for a product built from the transformed prime factors. The complete calculation is presented for the prime p = 2, with exact fractions and decimal values: the transformed coordinate is p₂₀ = 2 097 152/1 048 575 ≈ 2.0000019073504518, a displacement of Δp = 2/1 048 575 ≈ +1.9074 × 10⁻⁶. The reciprocal coordinate used in the Euler product moves from 1/2 to 1 048 575/2 097 152 ≈ 0.4999995231628418, a displacement of Δx₂ = −1/2 097 152 ≈ −4.7684 × 10⁻⁷. The paper then defines the transformed function ζ*₂₀(s) = ∏ₚ [1 − (cp)⁻ˢ]⁻¹ for Re(s) > 1, writes the corresponding transformed factors for the primes 17 and 19, and describes how these factors are evaluated on the critical line through complex powers. A figure shows the original and transformed positions of the prime 2 and of its reciprocal, with the displacements on a magnified scale. The paper states that any consequence for the location of the non-trivial zeros of the Riemann zeta function would require determining the zeros of the transformed function and relating them to those of ζ(s) through a demonstrated identity. This is left as the next stage of the investigation.