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Considering $p_{n}$ the nth prime number, then compute the limit:

$$\lim_{n\to\infty} \left\{ \dfrac{1}{p_{1}} + \frac{1}{p_{2}}+\cdots+\frac{1}{p_{n}} \right\} - \{\log{\log n } \}$$ where $\{ x \}$ denotes the fractional part of $x$.

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    Do you know that the series diverges like $\log \log n$, for starters? That is $$\sum_{n=1}^n \frac 1 {p_n} \sim \log \log n$$2012-06-15
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    @PeterTamaroff That doesn't really help though. It doesn't even show existence of the limit, e.g. $2^n$ diverges like $2^n+n$.2012-06-15
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    The limit does exist. However, to get an estimate of the limit you need to prove prime number theorem i.e. for this case we need to prove that $\psi(x) \sim x$.2012-06-15
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    @AlexBecker I'm talking about $o(1)$ here.2012-06-15

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