The Limiting Principle Expression of Prime Distribution and the Relativity of Constants

This paper establishes an exact correspondence between the prime distribution function π(x) and a class of nonlinear discrete dynamical systems. The core iteration system is given by [1]:

u_{n+1} = u_n^{1/s} e^{-π u_n / s^2}, s ∈ C \ {0,1}

For almost all initial values, this system converges to a unique fixed point u_s satisfying the balance equation [2]:

ln u_s / u_s = -π / (s(s-1)) (1)

The closed-form solution for u_s in terms of s is:

u_s = s(s-1)/π · W₀( π/(s(s-1)) ) (2)

where W₀ is the principal branch of the Lambert W function [3]. From equation (1), we derive the expression for s in terms of u_s:

s = (1 ± √(1 - 4π u_s/ln u_s)) / 2 (3)

A triple-space structure is introduced:

· U-space: u_{n+1} = u_n^{1/s} e^{-π u_n / s^2}, fixed point u_s

· Z-space: z_{n+1} = (z_n + e^{-π z_n / s})/s, fixed point z_s = s/π · W₀(π/(s(s-1)))

· V-space: v_{n+1} = v_n^{1/s} e^{π/(s^2 v_n)}, fixed point v_s = 1/u_s

These spaces satisfy the duality relations:

u_s · v_s = 1, u_s = e^{-π z_s / s}, v_s = e^{π z_s / s}

The main discovery of this paper is that when the parameter takes the special value s = 1/2 ± i√3/2, we have the exact equality:

1/π_s = x / ln x

where π_s = -s(s-1) · ln u_s / u_s. The Prime Number Theorem π(x) ∼ x/ln x [4,5,6] emerges as the asymptotic manifestation of this equality as s approaches this limiting point.

Equation (1) is a fundamental self-referential equation for exponential-logarithmic structures, exhibiting the relativity of constants: fixing different parameters (π, u, or s) yields different perspectives of the same underlying relationship. Each parameter is irreplaceable, forming a self-referential closed loop.