# A certain product over primes

There is this product over primes I came across, and I was wondering what the value would be asymptotically as $n$ goes to infinity. Could someone please help me out? Thank you! $$\prod_{\text{primes } p<n}\log n /\log p$$

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Looks a lot like $\to \infty$. For example with $n= 1000000$ I obtain $\approx 10^{2970}$. –  Hagen von Eitzen Nov 30 '12 at 18:54
Thank you very much for the computation! –  J Kasahara Nov 30 '12 at 23:06

The product diverges to $\infty$. Note that the factor is at least $2$ for primes $p<\sqrt n$; therefore the product is at least $2^{\pi(\sqrt{n})}$, which definitely tends to $\infty$ with $n$.
A more careful argument (taking the logarithm of the product and applying partial summation and the prime number theorem) shows that the product goes to infinity like $e^{n/\log^2n}$.