Asymptotic Formula for Totient Sum Over Shifted Primes with \(6/\pi^2\) Constant — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Asymptotic formula for sum of Euler totient over shifted primes reveals a constant term involving \(6/\pi^2\) and a \(\log\log x\) growth. | 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 \(6/\pi^2 \approx 0.607927\) (reciprocal of \(\zeta(2)\)). | CONNECTION: The constant \(6/\pi^2\) is the density of squarefree integers, linked to the hexagonal lattice packing density in 2D (circle packing ratio \(\pi/(2\sqrt{3}) \approx 0.9069\) is distinct, but \(6/\pi^2\) appears in lattice point counting for coprime pairs). No direct golden ratio or base-60 link. | DEPTH: 6 — solid analytic number theory result with a clean constant, but not a paradigm shift. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7170055028
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