Estimates of the minimum of the Gamma function using the Lagrange inversion theorem and the Fa\`a di Bruno formula

TL;DR

This paper derives an expansion for the minimum of the Gamma function using advanced mathematical tools, connecting it to the Riemann zeta function and providing series convergence insights.

Contribution

It introduces a novel series expansion for the Gamma function's minimum using the Lagrange inversion theorem and Faà di Bruno formula, linking it to the zeta function.

Findings

Series expansion up to γ^6 for convergence analysis

More accurate approximations at 3/2 using Lagrange inversion

Explicit connection between derivatives of digamma and zeta functions

Abstract

In this article we derive, using the Lagrange inversion theorem and applying twice the Fa\`a di Bruno formula, an expression of the minimum of the Gamma function $\Gamma$ as an expansion in powers of the Euler-Mascheroni constant $\gamma$. The result can be expressed in terms of values the Riemann zeta function $\zeta$ of integer arguments, since the multiple derivative of the digamma function $\psi$ evaluated in $1$ is precisely proportional to the zeta function. The first terms (up to $\gamma^6$) 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.