Difference between revisions of "Euler series"
From Encyclopedia of Mathematics
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$$\sum_p\frac1p,$$ | $$\sum_p\frac1p,$$ | ||
| − | 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 | + | 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 |
| − | $$\sum_{p\leq x}\frac1p=\ln\ln x+C+O\left(\frac{1}{\ln x}\right),$$ | + | $$\sum_{p\leq x}\frac1p = \ln\ln x+C+O\left(\frac{1}{\ln x}\right),$$ |
where $C=0.261497\ldots$. | where $C=0.261497\ldots$. | ||
| − | + | For a derivation of this asymptotic relation, see {{Cite|a1}}, Chap. 22.7, 22.8. | |
| − | |||
| − | |||
| − | For a derivation of | ||
====References==== | ====References==== | ||
| − | + | * {{Ref|a1}} G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Oxford Univ. Press (1979) {{ZBL|0423.10001}} | |
[[Category:Number theory]] | [[Category:Number theory]] | ||
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=53760
Euler series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_series&oldid=53760
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article