Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $\varphi(p_n-1)/(p_n-1)$.

share|cite|improve this question
up vote 2 down vote accepted

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

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.