Does an Extension of the Primorial Function Exist?

[quinn]

I only am wondering about the Primorial function, n#, (product of all primes less than or equal to n)

The gamma/factorial function has a nice recursive relationship that is composed of elementary functions; does there exsist an extension to the primorial function?

 

[shmoe]

There's a simple asymptotic for it's logarithm given by one form of the prime number theorem, log(n#)~n.

 

[Karlisbad]

If you define the Chebyshev function:

$$\theta (x)= \sum_{p<x} log(p)$$ then:

$$\theta (p_{n}) = log(p#)$$ but using this definition the PNT gives

$$log(p # ) \sim nlogn$$

 

[CRGreathouse]

p# is about $e^p$. Pierre Dusart has a paper with fairly tight bounds for this and other functions relating to prime counting.