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Difference between revisions of "Euler constant"

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The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036420/e0364201.png" /> defined by
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The number $\gamma$ defined by
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036420/e0364202.png" /></td> </tr></table>
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$$ \gamma=\lim_{n\to \infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n\right)\approx 0.57721566490\ldots,$$
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036420/e0364203.png" /></td> </tr></table>
 
  
 
considered by L. Euler (1740). Its existence follows from the fact that the sequence
 
considered by L. Euler (1740). Its existence follows from the fact that the sequence
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036420/e0364204.png" /></td> </tr></table>
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$$ 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln(n+1)$$
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is monotone increasing and bounded from above. The number-theoretic nature of the Euler constant has not been studied; it is not even known (1988) whether it is a rational number or not.
 
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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.
  
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====Comments====
 
 
In fact, a relation
 
In fact, a relation
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036420/e0364205.png" /></td> </tr></table>
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$$ \sum_{n\leq x}\frac{1}{n}-\ln x=\gamma+O\left(\frac{1}{x}\right$$
  
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holds, cf. [[#References|[a1]]], Chapt. 22.5.
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holds, cf. {{Cite|HaWr|Chapter 22.5}}.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapts. 5; 7; 8</TD></TR></table>
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|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
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Revision as of 20:06, 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=25611
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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