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So if using only Chebyshev's and Mertens's estimation, how to evaluate the following series: $$\sum_{p\geq 3}\frac{1}{p \log{\log{p}}} ?$$

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You would make life much easier for prospective answerers by presenting, or at least linking to, the estimations to which you refer. – Gerry Myerson Dec 11 '11 at 22:31
    
Basically $\psi(x) \sim x$ and the four Mertens type estimation can be found in math.uiuc.edu/~hildebr/ant/main3.pdf – Rob Dec 11 '11 at 22:38
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$\psi(x)\sim x$ is the Prime Number Theorem, which is stronger than Chebyshev. – Gerry Myerson Dec 12 '11 at 0:39

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