What's the approximation for $\prod_{p\leq n^2} p^{2n}$?

Question

I have 2 questions ($p$ prime):

1) I know that

$$\underset{p\leq n}{\prod}p^{\frac{1}{p-1}}\sim n$$

Does that mean

$$\underset{p\leq n^2}{\prod}p^{\frac{1}{p-1}}\sim n^2$$?

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2) What's the approximation for

$$\prod_{p\leq n^2} p^{2n}$$?

Thanks!

Answer 1

For the first question, if $f(n)\sim n$ then $f(n^2)\sim n^2.$

For the second, $$ \prod_{p\le x}p=e^{x+o(1)} $$ so $$ \prod_{p\le n^2}p^{2n}=\left(\prod_{p\le n^2}p\right)^{2n}=\left(e^{n^2+o(1)}\right)^{2n}=e^{2n^3+o(n)}. $$