The primorial $p_n\# $ is defined as the product of the first $n$ primes: $$p_n\# = \prod_{k = 1}^n p_k.$$ Asymptotically, primorials grow like $$p_n\# = e^{(1 + o(1))n\ln n)}.$$ How does one derive this asymptotic formula?
How to show how primorials grow asymptotically?
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2I believe this is equivalent to the Prime Number Theorem. If you want a proof *of* the Prime Number Theorem, this website is too small to contain one. Would you settle for a proof *from* the Prime Number Theorem? Also, have you looked into that book I mentioned in the comments on your earlier question? – 2012-11-18
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1@GerryMyerson Again, this is not the answer I am looking for. Apostol? I will. I believe telling someone to read a book is utterly pointless, because any question can be answers in this manner and I do not think people come here to learn about good books. They come here for answers. – 2012-11-18
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1Couldn't you have just said, "Yes, I'll settle for a proof from the Prime Number Theorem" (which is what you have done, by accepting the answer by @Jeff)? Don't worry about it, glebovg. You have convinced me not to try to help you. Ever. – 2012-11-18
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0I do not see your answer anywhere. There is only one answer, so I can only accept that answer. I was looking for a derivation, not some random claim. Yes, the derivation employs PNT, but the statement about primorials is not equivalent to PNT. – 2012-11-18
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0Did you not see that I asked you, "Would you settle for a proof from the Prime Number Theorem?"? – 2012-11-18
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0I think I misinterpreted your comment. I did not realize you were offering to post an answer. I am sorry. You may post a similar, but perhaps easier derivation and I will accept your answer instead. – 2012-11-18
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0I don't think I can improve on the answer posted by @Jeff. – 2012-11-18
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0OK. In any case, thank you. – 2012-11-18
1 Answers
Let $\pi(x)$ denote the prime counting function. Let $x\geq 1$. We will compute $\displaystyle\sum_{p\leq x\;prime} (\ln x - \ln p)$ where the sum is taken over all primes $p\leq x$. Note that
$\displaystyle\sum_{p\leq x\;prime} (\ln x - \ln p)=\displaystyle\sum_{p\leq x\;prime}\int^x_p 1/t dt=\int_1^x \frac{\pi(t)}{t}dt$.
Since there exists a constant $c$ such that $\pi(t)\leq \frac{ct}{\ln t}$, it follows that
$\displaystyle\sum_{p\leq x\;prime} (\ln x - \ln p)=O(\int_1^x \frac{1}{\ln t} dt)=o(x)$.
The prime number theorem gives
$\displaystyle\sum_{p\leq x\;prime}\ln p=\pi(x)\ln x - \displaystyle\sum_{p\leq x\;prime} (\ln x - \ln p)= \ln x (\frac{x}{ln x}+o(\frac{x}{ln x}))-o(x)=x+o(x)$.
Then $\displaystyle\sum_{k=1}^n \ln p_k = \displaystyle\sum_{p\leq p_n\;prime}\ln p=p_n+o(p_n)$. It is known that $p_n=n\ln n +o(n\ln n)$; see for example the Wikipedia article on the prime number theorem. Hence $\displaystyle\sum_{k=1}^n \ln p_k =n\ln n + o(n\ln n)$. Taking exponentials of both sides gives the result you asked for.