Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
    
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

1 Answer 1

up vote 2 down vote accepted

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

share|improve this answer
    
Thank you very much! –  J Kasahara Nov 30 '12 at 23:06

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.