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Where pn is the nth prime number, does the infinite product $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)$$ converge to a nonzero value? (Any help would be much appreciated!)

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  • $\begingroup$ What does the few expanded product to series look like? $\endgroup$
    – jimjim
    Oct 16, 2014 at 10:38
  • $\begingroup$ Since $ \left(1-\frac{1}{p_n}\right) \lt 1 $ we have that $$\prod_{k=1}^{n}\left(1-\frac{1}{p_k}\right) \lt 1 \;\; \forall n \in \Bbb N$$ $\endgroup$
    – Ishfaaq
    Oct 16, 2014 at 10:41
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    $\begingroup$ In case of infinite products they do not converge to zero, they diverge to zero $\endgroup$
    – jimjim
    Oct 16, 2014 at 10:42
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    $\begingroup$ Arjang and @Ishfaaq: infinite products all of whose terms are between zero and one CAN converge to non-zero values. See math.stackexchange.com/questions/141705/… for an example... What I am asking here is if $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)>0$$ But, as of Liu Gang's answer, it appears that $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)=0 \text{ since } \sum_{n=1}^{\infty}\frac{1}{p_n} diverges $$ so it does in fact converges to zero $\endgroup$
    – ManRow
    Oct 16, 2014 at 10:44
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    $\begingroup$ @ManRow : I was aware of that example, there was a comment that contained "convergence to 0" in case of infinite products that is no correct, they only diverge to 0. $\endgroup$
    – jimjim
    Oct 16, 2014 at 10:53

1 Answer 1

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If all $a_n \in (0,1)$, $\displaystyle\prod_{n=1}^{+\infty} (1- a_n)$ is non-zero if and only if $\sum_{n=1}^{+\infty} a_n < +\infty$

And we know that $\displaystyle\sum_{n=1}^{+\infty} \frac{1}{p_n} = +\infty$, you can find proofs here

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