Transcendence measure of e^1/n

2023/03/09 by Marta Dujella, Dujella, Marta, Anne-Maria Ernvall-Hytönen +5
Mathematics · #FOS: Mathematics #Functional Equations Stability Results #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2303.05542

openalex publication_date 2023/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a given transcendental number ξ and for any polynomial P(X)=: λ₀+⋯+λ_k X^k ∈ ℤ[X], we know that P(ξ) ≠ 0. Let k ≥ 1 and ω(k, H) be the infimum of the numbers r > 0 satisfying the estimate |λ₀+λ₁ ξ+λ₂ ξ²+ … +λ_kξ^k| gt; (1)/(H^r), for all (λ₀, … ,λ_k)^T ∈ ℤ^k+1∖\0\ with max_{1≤ i≤ k} \|λ_i|\ ≤ H. Any function greater than or equal to ω(k, H) is a \it transcendence measure of ξ. In this article, we find out a transcendence measure of e^1/n which improves a result proved by Mahler(\citeMahler) in 1975.

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