Difference between revisions of "Euler series"
From Encyclopedia of Mathematics
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The expression | The expression | ||
| − | + | $$\sum_p\frac1p,$$ | |
| − | where the sum extends over all prime number | + | 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 | + | 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=11331
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