Relation from Infinite Products to Infinite Sums

Question

I was messing with some products and found that $$\prod_{{n=1}}^{\infty} 2^{\frac{1}{n^2}} = 2^{\frac{\pi^2}{6}}$$

The index is equal to $\zeta(2)$. I can inductively see that the index will end up being the associated value of the zeta function, but why? What is that relation?

Answer 1

Just to put the solution in an answer,

$$ 2^x 2^y \dots 2^z = 2^{x+y+\dots +z} $$

so

$$ \prod 2^{\frac{1}{n^2}} = 2^{\sum \frac{1}{n^2}} = 2^\frac{\pi^2}{6} $$

Answer 2

Actually, I can see why this is the case now, my apologies. Right as I posted, I could see it.

The product is equal to $$2^{ \sum_{{n=1}}^{\infty} \frac{1}{n^2}}$$