Euler's Totient Density and the Probability of Coprime Integers — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The summatory Euler totient function's asymptotic density yields the probability 6/π² that two random integers are coprime, linking number theory to lattice visibility and geometric probability. | MATH: φ(n) = count of integers ≤ n coprime to n; summatory Φ(x) = Σ_{n≤x} φ(n) ~ (3/π²)x². Probability gcd(a,b)=1 = 1/ζ(2) = 6/π² ≈ 0.6079. Error term refined in arXiv:1206.2792v5: Φ(x) = (3/π²)x² + O(x log^(2/3)(x) (log log x)^(4/3)) — sharper than classical O(x log x). Reciprocal summatory: Σ_{n≤x} 1/φ(n) ~ C log x + O(1) with C = ζ(2)ζ(3)/ζ(6) ≈ 1.9436. | CONNECTION: 6/π² = 0.6079 — not a golden ratio, but note 1 − 6/π² = 0.3921 ≈ 0.382 (complement of golden ratio conjugate, within 2.6%). The lattice point visibility interpretation: from origin, fraction of integer lattice points visible = 6/π² — a crystallographic density in Z². The error term's exponent 2/3 and 4/3 relate to divisor problem geometry. Base-60 link: 6/π² × 60 = 36.47 ≈ 36 = 60 × 0.6, hinting at sexagesimal harmoni
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026eulers,
title = {{Euler's Totient Density and the Probability of Coprime Integers — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23446973},
url = {https://zenodo.org/records/23446973},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euler's Totient Density and the Probability of Coprime Integers — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23446973
UR - https://zenodo.org/records/23446973
AB - FINDING: The summatory Euler totient function's asymptotic density yields the probability 6/π² that two random integers are coprime, linking number theory to lattice visibility and geometric probability. | MATH: φ(n) = count of integers ≤ n coprime to n; summatory Φ(x) = Σ_{n≤x} φ(n) ~ (3/π²)x². Probability gcd(a,b)=1 = 1/ζ(2) = 6/π² ≈ 0.6079. Error term refined in arXiv:1206.2792v5: Φ(x) = (3/π²)x² + O(x log^(2/3)(x) (log log x)^(4/3)) — sharper than classical O(x log x). Reciprocal summatory: Σ_{n≤x} 1/φ(n) ~ C log x + O(1) with C = ζ(2)ζ(3)/ζ(6) ≈ 1.9436. | CONNECTION: 6/π² = 0.6079 — not a golden ratio, but note 1 − 6/π² = 0.3921 ≈ 0.382 (complement of golden ratio conjugate, within 2.6%). The lattice point visibility interpretation: from origin, fraction of integer lattice points visible = 6/π² — a crystallographic density in Z². The error term's exponent 2/3 and 4/3 relate to divisor problem geometry. Base-60 link: 6/π² × 60 = 36.47 ≈ 36 = 60 × 0.6, hinting at sexagesimal harmoni Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -