Estimates of the minimum of the Gamma function using the Lagrange inversion theorem and the Faà di Bruno formula
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910685744594944 |
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| author | Pain, Jean-Christophe |
| author_facet | Pain, Jean-Christophe |
| contents | In this article we derive, using the Lagrange inversion theorem and applying twice the Faà di Bruno formula, an expression of the minimum of the Gamma function $Γ$ as an expansion in powers of the Euler-Mascheroni constant $γ$. The result can be expressed in terms of values the Riemann zeta function $ζ$ of integer arguments, since the multiple derivative of the digamma function $ψ$ evaluated in $1$ is precisely proportional to the zeta function. The first terms (up to $γ^6$) were provided in order to address the convergence of the series. Applying the Lagrange inversion theorem at the value $3/2$ yields more accurate results, although less elegant formulas, in particular because the digamma function evaluated in $3/2$ does not simplify. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimates of the minimum of the Gamma function using the Lagrange inversion theorem and the Faà di Bruno formula Pain, Jean-Christophe Number Theory In this article we derive, using the Lagrange inversion theorem and applying twice the Faà di Bruno formula, an expression of the minimum of the Gamma function $Γ$ as an expansion in powers of the Euler-Mascheroni constant $γ$. The result can be expressed in terms of values the Riemann zeta function $ζ$ of integer arguments, since the multiple derivative of the digamma function $ψ$ evaluated in $1$ is precisely proportional to the zeta function. The first terms (up to $γ^6$) were provided in order to address the convergence of the series. Applying the Lagrange inversion theorem at the value $3/2$ yields more accurate results, although less elegant formulas, in particular because the digamma function evaluated in $3/2$ does not simplify. |
| title | Estimates of the minimum of the Gamma function using the Lagrange inversion theorem and the Faà di Bruno formula |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.03181 |