View source for Euler series

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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 number]]s is infinite. The partial sums of the Euler series satisfy the asymptotic relation

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

where $C=0.261497\ldots$.

For a derivation of this asymptotic relation, see {{Cite|a1}}, Chap. 22.7, 22.8.

====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]]