# How to show how primorials grow asymptotically?

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?

• I 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? Nov 18, 2012 at 0:32
• @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. Nov 18, 2012 at 1:09
• Couldn'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. Nov 18, 2012 at 4:39
• I 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. Nov 18, 2012 at 5:07
• Did you not see that I asked you, "Would you settle for a proof from the Prime Number Theorem?"? Nov 18, 2012 at 5:57

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.