Artin's Conjecture: Density Formula via Prime Local Factors — E8 Intelligence Research

FINDING: Artin's primitive root conjecture concerns the density of primes for which a fixed integer is a primitive root; the generalized form yields an asymptotic formula with a product over primes of local factors. | MATH: For a fixed integer \\(a \\neq -1, \\square\\), the conjectured density is \\(C_a = \\prod_{p \\mid a} \\left(1 - \\frac{1}{p(p-1)}\\right) \\prod_{p \\nmid a} \\left(1 - \\frac{1}{p(p-1)}\\right)^{-1}\\) — but the cleanest form is \\(C_a = \\prod_{p} \\left(1 - \\frac{1}{p(p-1)}\\right) \\cdot \\prod_{p \\mid a} \\frac{p(p-1)}{p^2 - p - 1}\\). The generalized version (arXiv:1504.00843v9) gives an asymptotic for the count of \\(n \\le x\\) such that \\(a\\) is a primitive root modulo \\(n\\), with main term \\(x \\cdot C_a' + O(x^{1-\\delta})\\). | CONNECTION: The local factor \\(1 - \\frac{1}{p(p-1)}\\) has no direct 0.618/1.618 ratio, but the product structure mirrors the Euler product for the zeta function — a multiplicative lattice over primes. The density constants are not simple algebraic ratios; th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com