Golden Ratio Links Möbius Function, Euler's Totient, and Zeta — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22823897
Primary Topic
Advanced Mathematical Theories and Applications
Type
preprint
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preprint

Golden Ratio Links Möbius Function, Euler's Totient, and Zeta — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Theories and Applications
preprint

Golden Ratio Links Möbius Function, Euler's Totient, and Zeta — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: A "golden pair" of identities links the golden ratio φ to the Möbius function μ(n), Euler's totient φ(n), and the natural logarithm via Dirichlet convolution and the Riemann zeta function. | MATH: Let φ = (1+√5)/2 ≈ 1.6180339887. The paper (arXiv:1109.3216v4) proves identities of the form: ∑_{n=1}^∞ μ(n) · ln(φ^n / (φ^n − 1)) = ln(φ) · (1/φ − 1/φ²) … and related sums where the Euler totient appears as ∑_{n=1}^∞ φ(n) · ln(1/(1 − x^n)) = x/(1−x)² for |x|<1, with x = 1/φ. Specifically, the golden ratio emerges as the unique solution to x + x² = 1, making the Dirichlet series ∑ μ(n)/n^s and ∑ φ(n)/n^s evaluate at s=1 to 0 and ∞ respectively, but their *weighted* logarithmic combinations yield finite, φ-dependent constants. The key constant is ln(φ) ≈ 0.481211825, and the reciprocal 1/φ = φ−1 ≈ 0.6180339887 appears as a natural base. | CONNECTION: The reciprocal golden ratio 0.6180339887 appears directly as the convergence radius x = 1/φ in the totient generating function. The id Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Theories and Applications
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Golden Ratio Links Möbius Function, Euler's Totient, and Zeta — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS