On the Uniqueness of Ein(1) among Linear Combinations of the Euler-Mascheroni and Euler-Gompertz Constants
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911187531202560 |
|---|---|
| author | Powers, Michael R. |
| author_facet | Powers, Michael R. |
| contents | From a well-known equation of Hardy, one can derive a simple linear combination of the Euler-Mascheroni constant ($γ=0.577215\ldots$) and Euler-Gompertz constant ($δ=0.596347\ldots$): $γ+δ/e=\textrm{Ein}\left(1\right)$. Although neither $γ$ nor $δ$ is currently known to be irrational, this linear combination has been shown to be transcendental (by virtue of the fact that it appears as an algebraic point value of a particular E-function). Moreover, both pairs ($γ$,$δ$) and ($γ$,$δ/e$) are known to be disjunctively transcendental. In light of these observations, we investigate the impact of the coefficient $α$ in combinations of the form $γ+αδ$, and find that $α=1/e$ is the unique coefficient value such that canonical Borel-summable divergent series for $γ$ and $δ$ can be linearly combined to force conventional convergence of the resulting series. We further indicate how this uniqueness property extends to a sequence of generalized linear combinations, $γ^{\left(n\right)}+αδ^{\left(n\right)}$, with $γ^{\left(n\right)}$ and $δ^{\left(n\right)}$ given by (ordinary and conditional) moments of the Gumbel(0,1) probability distribution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00315 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Uniqueness of Ein(1) among Linear Combinations of the Euler-Mascheroni and Euler-Gompertz Constants Powers, Michael R. Number Theory Classical Analysis and ODEs Complex Variables 11J85, 11J81 From a well-known equation of Hardy, one can derive a simple linear combination of the Euler-Mascheroni constant ($γ=0.577215\ldots$) and Euler-Gompertz constant ($δ=0.596347\ldots$): $γ+δ/e=\textrm{Ein}\left(1\right)$. Although neither $γ$ nor $δ$ is currently known to be irrational, this linear combination has been shown to be transcendental (by virtue of the fact that it appears as an algebraic point value of a particular E-function). Moreover, both pairs ($γ$,$δ$) and ($γ$,$δ/e$) are known to be disjunctively transcendental. In light of these observations, we investigate the impact of the coefficient $α$ in combinations of the form $γ+αδ$, and find that $α=1/e$ is the unique coefficient value such that canonical Borel-summable divergent series for $γ$ and $δ$ can be linearly combined to force conventional convergence of the resulting series. We further indicate how this uniqueness property extends to a sequence of generalized linear combinations, $γ^{\left(n\right)}+αδ^{\left(n\right)}$, with $γ^{\left(n\right)}$ and $δ^{\left(n\right)}$ given by (ordinary and conditional) moments of the Gumbel(0,1) probability distribution. |
| title | On the Uniqueness of Ein(1) among Linear Combinations of the Euler-Mascheroni and Euler-Gompertz Constants |
| topic | Number Theory Classical Analysis and ODEs Complex Variables 11J85, 11J81 |
| url | https://arxiv.org/abs/2510.00315 |