Finding an upperbound for $\sum\limits_{i=2}^{n}\bigg(\prod\limits_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$

Question

I was wondering whether there exists a known upperbound for:

$$f(n)=\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$$

For example:

$$f(4)=\dfrac{1}{3}+\dfrac{1\cdot3}{3\cdot5}+\dfrac{1\cdot3\cdot5}{3\cdot5\cdot7}+\dfrac{1\cdot3\cdot5\cdot9}{3\cdot5\cdot7\cdot11}$$

I've searched around for a bit, but since english is not my native language, I've been unable to phrase this question in a way that google understands.

I'm really hoping for something in terms of $\log(n)$ or better.

Any kind of help is really appreciated.

Answer 1

Well $$\prod_{k=2}^{n} \frac{p_{k}-2}{p_{k}}<\prod_{k=2}^{n} \frac{p_{k}-1}{p_{k}}=2\prod_{k=1}^{n} \frac{p_{k}-1}{p_{k}}\sim \frac{2e^{-\gamma }}{\ln{n}}$$

This means that there $\exists \varepsilon>0$, constant such that $$\prod_{k=2}^{n} \frac{p_{k}-2}{p_{k}} < (1+\varepsilon) \frac{2e^{-\gamma }}{\ln{n}}$$ always.

For more details see Mertens' theorems. Or section 22.8 of this book.