Polylogarithm Values at a Golden Ratio-Based Argument

Description

The research paper by Tristen Harr introduces and analyzes a novel complex constant, ΛG1, which is derived from inverse powers of the golden ratio, ϕ. The author defines the constant as ΛG1=T+iJ, where T=1/(2ϕ) and J=1/(2ϕ2), and proves it is an algebraic number with a magnitude less than one. This property validates its use as an argument in the Polylogarithm function, Lis(z). Based on high-precision numerical evaluations for the Dilogarithm (s=2) and Trilogarithm (s=3) cases, the paper conjectures that the resulting values, Lis(ΛG1), are transcendental for all integers s≥2 and do not exist within the field extension Q(π,ln(2),ϕ). This investigation is partly motivated by potential applications in the study of quasicrystals, where the golden ratio plays a foundational role.

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