On the Algebraic Independence of a Set of Generalized Constants

2025/08/16 by Powers, Michael R. · 1 citation
#11J81 #11J85 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.12123

Abstract

Neither the Euler-Mascheroni constant, γ=0.577215…, nor the Euler-Gompertz constant, δ=0.596347…, is currently known to be irrational. However, it has been proved that these two numbers are disjunctively transcendental; that is, at least one of them must be transcendental. The two constants are related through a well-known equation of Hardy, which recently has been generalized to a pair of infinite sequences (γ⁽ⁿ⁾,δ⁽ⁿ⁾) based on moments of the Gumbel(0,1) probability distribution. In the present work, we demonstrate the algebraic independence of the set \γ⁽ⁿ⁾+δ⁽ⁿ⁾/e\_n≥0, and thus the transcendence of γ⁽ⁿ⁾+δ⁽ⁿ⁾/e for all n≥0. This further implies the disjunctive transcendence of both pairs (γ⁽ⁿ⁾,δ⁽ⁿ⁾/e) and (γ⁽ⁿ⁾,δ⁽ⁿ⁾) for all n≥1.

Citations

• On linear forms containing the Euler constant
• Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms
• A refined version of the Siegel-Shidlovskii theorem

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