Inequalities and asymptotics for the Euler–Mascheroni constant based on DeTemple’s result

Abstract

Let $R_{n}={\sum }_{k=1}^{n}\frac {1}{k}-\ln \left (n+\frac {1}{2}\right )$. DeTemple proved the following inequality:

$\frac {1}{24(n+1)^{2}}<R_{n}-\gamma <\frac {1}{24n^{2}}$

for all integers n ≥ 1, where γ denotes the Euler–Mascheroni constant. In this paper, we give a pair of recurrence relations for determining the constants a _ℓ and b _ℓ such that

$R_{n}-\gamma \sim \sum\limits_{\ell =1}^{\infty }\frac {a_{\ell }}{(n^{2}+n+b_{\ell })^{2\ell -1}},\qquad n\to \infty .$

Based on this expansion, we establish some inequalities for the Euler–Mascheroni constant.