Difference between revisions of "Euler constant"
From Encyclopedia of Mathematics
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The number $\gamma$ defined by | The number $\gamma$ defined by | ||
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| − | $$ \sum_{n\leq x}\frac{1}{n}-\ln x=\gamma+O\left(\frac{1}{x}\right$$ | + | $$ \sum_{n\leq x}\frac{1}{n}-\ln x=\gamma+O\left(\frac{1}{x}\right)$$ |
holds, cf. {{Cite|HaWr|Chapter 22.5}}. | holds, cf. {{Cite|HaWr|Chapter 22.5}}. | ||
Revision as of 20:07, 27 April 2012
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. [HaWr, Chapter 22.5].
References
| [HaWr] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 |
How to Cite This Entry:
Euler constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_constant&oldid=25612
Euler constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_constant&oldid=25612
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article