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Euler constant

From Encyclopedia of Mathematics
Revision as of 18:00, 31 October 2014 by Richard Pinch (talk | contribs) (Also known as the ''Euler-Mascheroni'' constant, cite Finch (2003))
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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 MR0568909 Zbl 0423.10001

Comments

Also known as the Euler-Mascheroni constant.

References

[Fi] Steven R. Finch, "Mathematical constants" , Encyclopedia of mathematics and its applications 94, Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001
How to Cite This Entry:
Euler constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_constant&oldid=33211
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article