What is the value of this Infinite Product of prime numbers expression?

Question

What is the value of:

$$\prod_1^\infty \frac{p_i^2}{p_i^2 -1 }$$ Where $$p_i$$ are the prime numbers: 2, 3, ...

Answer 1

The value is $\pi^2/6$.

Here are two steps to take (made edit to the equation):

• $\frac{p^2}{p^2-1} = \frac{1}{1 - p^{-2}}$

• $\frac{1}{1 - p^{-2}} = \sum_{j \ge 0} p^{-2j}$.

Now, taking the product of all these sums you get $\sum_{n \ge 1} n^{-2}$. For the last you likely know the value; if not I recalled it above.