Is there a asymptotic formula for product of primes?

Question

$$P(x)=\prod_{p\leq x}p$$

As you can see P(x) represents the product of primes which are not greater than x. Is there a asymptotic formula for this?

Answer 1

Note that if $\vartheta$ is the Chebyshev function, we have the relationship

\begin{align*} e^{\vartheta(x)} &= \operatorname{exp} \left(\sum_{p \le x} \log p\right) \\ &= \prod_{p \le x} p \end{align*}

So asymptotic bounds for the Chebyshev function carry over to the desired product.