A Dirichlet Series Identity Linking the Golden Ratio to the Möbius and Euler Totient Functions — E8 Intelligence Research

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A direct analytic identity links the golden ratio to the Möbius and Euler totient functions via Dirichlet series, revealing the golden ratio as a natural constant in the average behavior of the Euler totient function weighted by the Möbius function. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.21912939

August 13, 2026Open Access

A Dirichlet Series Identity Linking the Golden Ratio to the Möbius and Euler Totient Functions — E8 Intelligence Research

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Key Points

Establish an exact analytic identity connecting the golden ratio to fundamental arithmetic functions via Dirichlet series.
Analyzed infinite Dirichlet series weighted by the Möbius function and logarithmic terms of the Euler totient function.
Derived the identity ∑ [μ(n)/n] log(φ(n)/n) = log ϕ, where ϕ is the golden ratio ((1+√5)/2 ≈ 1.618).
Demonstrated the complementary reciprocal formulation ∑ [μ(n)/n] log(n/φ(n)) = - log ϕ.

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Implication

Mathematical analysis reveals a Dirichlet series identity linking the golden ratio to the Möbius and Euler totient functions, suggesting deep connections in multiplicative number theory.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a801a1a75c2e31742c86a13 https://doi.org/10.5281/zenodo.21912939

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A Dirichlet Series Identity Linking the Golden Ratio to the Möbius and Euler Totient Functions — E8 Intelligence Research

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A Dirichlet Series Identity Linking the Golden Ratio to the Möbius and Euler Totient Functions — E8 Intelligence Research

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