Euler series

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