Estimates of the minimum of the Gamma function using the Lagrange inversion theorem and the Faà di Bruno formula

Abstract

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.