Approximation formulas and inequalities for the Euler-Mascheroni constant

Abstract

Let $a\in (0, \infty )$ and $s \in (0, 1)$, and let $\gamma$ denote the Euler-Mascheroni constant. The sequence $\{y_n(a, s)\}_{n\in \mathbb {N}}$ is defined by

$$\begin{aligned} y_n(a, s)&=\frac{1}{a^s}+\frac{1}{(a+1)^s}+\cdots +\frac{1}{(a+n-1)^s}-\frac{(a+n-1)^{1-s}-a^{1-s}}{1-s} \end{aligned}$$

for each $n\in \mathbb {N}:=\{1,2,3,\ldots \}$. The sequence $\{y_n(a, s)\}_{n\in \mathbb {N}}$ is convergent and its limit, denoted by $\ell (a, s)$, is a generalized Euler-Mascheroni constant. In this paper, we consider complete asymptotic expansions and inequalities related to generalized Euler-Mascheroni constant $\ell (a,s)$. We determine the coefficients $a_j$ and $b_j$ such that

$$\begin{aligned} \sum _{k=1}^{n}\frac{1}{k}-\ln \frac{n^p+\sum _{j=1}^{p}a_jn^{p-j}}{n^q+\sum _{j=1}^{q}b_jn^{q-j}}=\gamma +O\left( \frac{1}{n^{p+q+1}} \right) ,\qquad n\rightarrow \infty , \end{aligned}$$

where $p\ge 1$ and $q\ge 0$ are any given integers and $p=q+1$. This solves an open problem of Mortici.