A certain product over primes

Question

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! $$ \prod_{\text{primes } p<n}\log n /\log p $$

Answer 1

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 $e^{n/\log^2n}$.