I am looking for the asymptotic growth of product of consecutive primes. Is there anything that is known about this growth?
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Denote by $$\Pi(x)=\prod_{p\leqslant x}p,$$ thus $$\log\Pi(x)=\sum_{p\leqslant x}\log p:=\theta(x)\sim x,$$ which is known as the Prime Number Theorem. You may find further information in http://en.wikipedia.org/wiki/Prime_number_theorem 


I think you're asking about the primorial function $n\sharp$, the product of all the primes less than or equal to $n$. This satisfies $ n \sharp = \exp( n(1+o(1)) ) $. 


You can also prove that $$\displaystyle\lim_n\left(\prod_1^n p_i\right)^{1/p_n} = e$$ (where $p_i$ is the $i$th prime number and $e$ is Euler's exponential number) 

