Inequalities and asymptotics for the Euler-Mascheroni constant based on DeTemple's result
The author considers the approximation of the Euler-Mascheroni constant \(\gamma\) as the limit of the difference between the harmonic series and the logarithm. In particular, DeTemple's inequalities: \[ {1\over24(n+1)^2}<\left[\sum_{k=1}^n{1\over k}-\ln\left(n+{1\over 2}\right)\right]-\gamma<{1\over24n^2},\quad n=1,2,3,\dots \] The author goes one step forward and shows that \[ \left[\sum_{k=1}^n{1\over k}-\ln\left(n+{1\over 2}\right)\right]-\gamma\sim\sum_{k=0}^\infty{a_k\over(n^2+n+b_k)^{2k-1}}, \] where the constants \(a_k\) and \(b_k\) are determined by means of certain nonlinear recurrence relations. Previously, the author has obtained a similar expansion for the digamma function. As a consequence, the following improvement of DeTemple's inequalities are given: \[ {1\over24(n^2+n+\alpha)}<\left[\sum_{k=1}^n{1\over k}-\ln\left(n+{1\over 2}\right)\right]-\gamma<{1\over24(n^2+n+\beta)}, \quad n=1,2,3,\dots \] for certain constants \(\alpha\) and \(\beta\). Another more elaborated couple of inequalities are also given.
scientific article; zbMATH DE number 6654601
scientific article