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Let $p$ be a prime number. Denote by $P$ the set of all primes which are not greater than $p$.

Is there a well known estimation of the product of all prime numbers in $P$ (i.e. $\prod_{q\in P}q$)?

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2 Answers 2

up vote 4 down vote accepted

The product of the first $n$ primes is called the $n$-th primorial: $$p_n \# = \prod_{k=1}^n p_n.$$

An estimate for their growth is $p_n\# =\exp((1+\mathcal{o}(1) )n\log n).$

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Thanks! That's exactly what I was looking for. –  Tom Jun 10 '12 at 15:43
    
@Tom No problem =). –  Ragib Zaman Jun 10 '12 at 15:43
    
Does anyone have a link for a proof of the aforementioned growth rate of the primorial? –  Tom Jun 10 '12 at 20:18
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@Tom The claim above is equivalent to the prime number theorem. You can google for proof of prime number theorem. There are two main proofs. The first main proof was by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896 which uses complex analysis. The second main proof was by Atle Selberg and Paul Erdős in 1949 and is an "elementary" proof. ("elementary" here denotes that the proof doesn't use complex analysis. The proof is supposedly much harder than any proof using complex analysis.) –  user17762 Jun 11 '12 at 0:53

What you are looking for is $\exp(\theta(x))$ where $\theta(x)$ is the first Chebyshev function. $$\theta(x) = \sum_{\underset{p \leq x}{p-\text{prime}}} \log(p)$$It is also related to the primorial, which is the product of the first $n$ primes.

The fact that $\theta(x) \sim x$ i.e. $\theta(x) = x + o(x)$ is equivalent to the prime number theorem. We can get a better quantification of $o(x)$ in-fact. This is obtained while proving the prime number theorem. We can get that $$\theta(x) = x + \mathcal{O}(x \exp(-c (\log x)^{\lambda}))$$ for some constant $c$ and $\lambda$.

Also, the fact that $$\theta(x) = x + \mathcal{O}(x^{1/2+\epsilon})$$ is equivalent to Riemann hypothesis, where the power $\epsilon$ takes into account some $\log$ factors inside it.

EDIT

I am adding this to answer Tom's comment above. The claim that $$p_n\# =\exp((1+\mathcal{o}(1) )n\log n)$$ is equivalent to the prime number theorem. You can google for proof of prime number theorem. There are two main proofs. The first main proof was by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896 which uses complex analysis. The second main proof was by Atle Selberg and Paul Erdős in 1949 and is an "elementary" proof. ("elementary" here denotes that the proof doesn't use complex analysis. However, the elementary proof is supposedly much harder than any proof using complex analysis.)

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Thanks @Marvis! –  Tom Jun 11 '12 at 5:21

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