I was wondering if anyone knew of good approximations of the primorial function for large numbers, or of reasonable bounds for it. By primorial I mean:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]n\#=\prod_{p\le n}p[/tex] forpprime.

All I know is [tex]n\#\sim e^n[/tex] and the trivial [tex]\pi(n)!\le n\#\le n![/tex].

For small numbers (n < 1,000,000,000), e^n is larger than n#. Does this hold generally?

Obviously, the crude upper limit can be hacked slightly: take out the evens from the factorial and multiply by 2, take out the multiples of 3 and divide by 3, etc. In the extreme case this is just calculating the primorial itself, so not very useful. The lower bound can be improved to

[tex]p_n\#\le n!\prod^n_{k=5}\ln k+\ln\ln k-1[/tex]

using lower bounds on the primes, but this is unwieldy, and I'm pretty sure [tex]\lim_{n\rightarrow\infty}\frac{p_n\#}{n!\Pi}=0[/tex] where the product is as above. Using the best bounds out there (by Dusart) improves the result slightly at the cost of complicating it further.

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# Primorial: asymptotics and bounds

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