Number Theory – Convergence of Infinite Product of Prime Reciprocals

convergence-divergenceinfinite-productlimitsnumber theoryprime numbers

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!)

Best Answer

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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