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The expression
 
The expression
  
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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/e036570/e0365701.png" /></td> </tr></table>
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$$\sum_p\frac1p,$$
  
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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 number]]s is infinite. The partial sums of the Euler series satisfy the asymptotic 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/e036570/e0365703.png" /></td> </tr></table>
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$$\sum_{p\leq x}\frac1p = \ln\ln x+C+O\left(\frac{1}{\ln x}\right),$$
  
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036570/e0365704.png" />.
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where $C=0.261497\ldots$.
  
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For a derivation of this asymptotic relation, see {{Cite|a1}}, Chap. 22.7, 22.8.
  
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====References====
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* {{Ref|a1}} G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Oxford Univ. Press (1979) {{ZBL|0423.10001}}
  
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====Comments====
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[[Category:Number theory]]
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For a derivation of the asymptotic relation above see [[#References|[a1]]], Chapts. 22.7, 22.8.
 
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====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>
 

Latest revision as of 15:21, 10 April 2023

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 $C=0.261497\ldots$.

For a derivation of this asymptotic relation, see [a1], Chap. 22.7, 22.8.

References

  • [a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Oxford Univ. Press (1979) Zbl 0423.10001
How to Cite This Entry:
Euler series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_series&oldid=11331
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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