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Do we know the limit of the product

$$\prod_{p\text{ prime}}\left(1-\frac{1}{p(p-1)}\right)$$

?

I ask because it seems to me on heuristic grounds (but I believe I could make them rigorous) that this number should be the average probability that a uniformly chosen element of $(\mathbb{Z}/p\mathbb{Z})^\times$ is a generator, i.e. the Cesaro mean of the sequence $\varphi(p_n-1)/(p_n-1)$.

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1 Answer 1

up vote 2 down vote accepted

That limit exists and it's called the Artin's constant:

http://mathworld.wolfram.com/ArtinsConstant.html

It's related to the Artin's conjecture about primitive roots, so your heuristic reasonin gs are quite correct ;)

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