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Difference between revisions of "Euler series"

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The expression
 
The expression
  
<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/e036570/e0365701.png" /></td> </tr></table>
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$$\sum_p\frac1p,$$
  
where the sum extends over all prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036570/e0365702.png" />. L. Euler (1748) showed that this series diverges, thus providing another proof of the fact that the set of prime numbers is infinite. The partial sums of the Euler series satisfy the asymptotic relation
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where the sum extends over all prime number $p$. L. Euler (1748) showed that this series diverges, thus providing another proof of the fact that the set of prime numbers is infinite. The partial sums of the Euler series satisfy the asymptotic relation
  
<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/e036570/e0365703.png" /></td> </tr></table>
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$$\sum_{p\leq x}\frac1p=\ln\ln x+C+O\left(\frac{1}{\ln x}\right),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036570/e0365704.png" />.
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where $C=0.261497\ldots$.
  
  

Revision as of 17:59, 30 July 2014

The expression

$$\sum_p\frac1p,$$

where the sum extends over all prime number $p$. L. Euler (1748) showed that this series diverges, thus providing another proof of the fact that the set of prime numbers is infinite. The partial sums of the Euler series satisfy the asymptotic relation

$$\sum_{p\leq x}\frac1p=\ln\ln x+C+O\left(\frac{1}{\ln x}\right),$$

where $C=0.261497\ldots$.


Comments

For a derivation of the asymptotic relation above see [a1], Chapts. 22.7, 22.8.

References

[a1] 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 series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_series&oldid=32600
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article