View source for Euler constant

{{TEX|done}}
The number $\gamma$ defined by

$$ \gamma=\lim_{n\to \infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n\right)\approx 0.57721566490\ldots,$$

considered by L. Euler (1740). Its existence follows from the fact that the sequence

$$ 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln(n+1)$$

is monotone increasing and bounded from above. The number-theoretic nature of the Euler constant has not been studied; it is not even known (2012) whether it is a rational number or not.

In fact, a relation

$$ \sum_{n\leq x}\,\frac{1}{n}-\ln x=\gamma+O\left(\frac{1}{x}\right)$$

holds, cf. {{Cite|HaWr|Chapter 22.5}}.

====References====
{|
|-
|valign="top"|{{Ref|HaWr}}||valign="top"|  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapts. 5; 7; 8 {{MR|0568909}} {{ZBL|0423.10001}}
|-
|}

====Comments====
Also known as the ''Euler-Mascheroni'' constant.

====References====
{|
|-
|valign="top"|{{Ref|Fi}}||valign="top"|  Steven R. Finch,   "Mathematical constants" , Encyclopedia of mathematics and its applications '''94''', Cambridge University Press  (2003)  ISBN 0-521-81805-2 Zbl 1054.00001
|-
|}