Euler product question

Question

im doing a small research and i want to know if

$$P = 2 \times 3 \times 5 \times 7 \times 11 \times \cdots \times P_n$$

can be obtained using the euler product (since $s = 1$ on euler product gives you all prime number total equal to sum of natural numbers).

Im using a finite set of primes, which means i need to know how can i get the product of all primes until $P_n$ using sums or other method.

Thanks!!

Answer 1

Taking logarithm you get $\sum \log p_n$, which is more or less one of the Chebyshev functions. Investigating this function, the Euler product enters the scene.

Anyway, these products are known as Primorials; and all kinds of things are known.