A Survey of Euler's Constant

Multiple-correction and continued fraction approximation

“…See e.g. [11,14,15,20,21,[29][30][31]39,49] and references therein. Let R 1 (n) = a 1 n and for k ≥ 2 Lu [29] introduced the continued fraction method to investigate this problem, and showed 1 120(n + 1) 4 < r 3 (n) − γ < .…”

“…Similarly, one may check [14], Havil [25] and Lagarias [26]. For example, a long-standing open problem is whether or not it is a rational number.…”

“…the survey papers or books of R.P. Brent and P. Zimmermann [4], Dence and Dence [14], Havil [25] and Lagarias [26]. For example, a longstanding open problem is whether or not it is a rational number.…”

A new sequence convergent to Euler–Mascheroni constant

“…The Euler–Mascheroni constant was first introduced by Leonhard Euler (1707–1783) in 1734 as the limit of the sequence There are many famous unsolved problems about the nature of this constant (see, e.g., the survey papers or books of Brent and Zimmermann [1], Dence and Dence [2], Havil [3], and Lagarias [4]). For example, it is a long-standing open problem if the Euler–Mascheroni constant is a rational number.…”

“…Up to now, many authors are preoccupied to improve its rate of convergence; see, for example, [2, 5–14] and references therein. We list some main results: …”