Euler series

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The expression


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.


  • [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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Euler series. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article