Asymptotic Sum of Totient Over Shifted Primes Yields 6/π² and log log x — E8 Intelligence Research

FINDING: Asymptotic sum of Euler totient function over shifted primes yields a constant involving \(6/\pi^2\) and a \(\log\log x\) term. | MATH: \(\sum_{p \leq x} \varphi([x/p]) = \frac{6}{\pi^2} x \log\log x + c_0 x + O\left(x (\log x)^{-1}\right)\); constant \(c_0\) involves Euler–Mascheroni and prime zeta contributions. | CONNECTION: The coefficient \(6/\pi^2 = 1/\zeta(2)\) is the density of squarefree integers, linked to the probability that two random integers are coprime — a fundamental ratio in lattice point visibility and crystallographic reciprocal space. No direct golden ratio or base-60 link. | DEPTH: 6 — solid analytic number theory result, but not a breakthrough in geometric harmony. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com