Factorial: product over primes

Question

Does

$$\prod_{n=1}^{x}p_{n}$$

($p_n$ is the $n^{th}$ prime) have a name or has it been been used in math? Does it matter? And Is there an analytical continuation of it?

Answer 1

It is called the primorial. http://mathworld.wolfram.com/PrimeProducts.html. It is used in the proof that there are an infinite number of primes ... Assume there is a finte number of primes $p_1,p_2, \cdots , p_N$, then consider \begin{eqnarray*} P= \prod_{i=1}^{N} p_i +1 \end{eqnarray*} This cannot be divisible by any of the previous primes ... contradicting the assumption that there are a finite number of primes.

Answer 2

Apart from the answer of Donald Splutterwit, as a complement for the remaining question, the closest expression similar to an analytic continuation I can recall is the first Chebyshev function:

$$\vartheta(x)=\sum_{p \le x} \log {p}$$

In essence, it is equal to the natural logarithm of the primorial of $x$, which is the product of all the existing primes up to $x$, and $x$ indeed can be any real number as long as the prime numbers are positive integers:

$$\vartheta(x)=\sum_{p \le x} \log {p}= \log{p_1} + \log{p_2} + \cdots + \log{p_n} = log {\ (p_1p_2\cdots\ p_n)} = \log {\prod_{n=1}^{x}p_n}$$

Thus:

$$\prod_{n=1}^{x}p_n=e^{\vartheta(x)}, x \in \Bbb R$$

Said that, I am not sure if this is really practical as a true analytic continuation or not, but it provides a continuation over $\Bbb R$. For instance Ramanujan used the Chebyshev functions to prove Bertrand's postulate, and for that proof he linked the Chebyshev functions applied to the problem, with Stirling's approximation and the Gamma function (extension of the factorial function), so in practice they seem close to an analytic extension.