With $p\_i$ being the $i$-th prime, I'm wondering whether there is a tighter bound than $\alpha = 4$ in the relation
$$
\prod\_{i=1}^n p\_i < \alpha^{p\_n}
$$
$\alpha = 4$, which is tight enough to be used to prove Bertrand's postulate (Chebyshev's theorem) that there is always a prime in the interval $[k,2k]$, is easily proven using $\binom{2r}{r} < 4^r$ and the fact that any prime in the interval $[n/2,n]$ will divide $\binom{2r}{r} < 4^r$; see the answer to 


<https://math.stackexchange.com/questions/1924453/show-that-p-1p-2-cdots-p-t4n>


However, that argument makes a couple of lavishly conservative statements, two of which are that since $p\_n$ divides $\binom{2n}{n}$ it must be less than $\binom{2n}{n}$, and that $\binom{2n}{n} < \sum\_k \binom{2n}{k}$. One might suspect that a smaller value of $\alpha$ could suffice.


Numerical experimentation finds that for $n \leq 100$, 


$$
\prod\_{i=1}^n p\_i < (2.62)^{p\_n}
$$
 but if you extend to $n=1000$, you need to increase to $\alpha = 2.6938$.


Thus my question: Is it known that no $\alpha < 4$ works in the inequality as $n\to \infty$, has there been a lower value demonstrated, or is this question completely open? 


I would guess that this can be attacked using the Prime Number Theorem, but fluctuations from that distribution become important for this bound. 



Is there a tighter bound than $\alpha=4$ in $ \prod_{i=1}^n p_i < \alpha^{p_n} $?
It follows from the Prime Number Theorem that the $p\_n$-th root of the product tends to $e=2.7182\dots$. In practice one takes the logarithm of the product and divides by $p\_n$, which then tends to $1$. This means that any $\alpha>e$ produces an upper bound for sufficiently large $n$, while any $\alpha<e$ produces a lower bound for sufficiently large $n$.


For good practical bounds see Rosser-Schoenfeld: Approximate formulas for some functions of prime numbers. See especially (3.15) and (3.16) there, which state in particular
$$1-\frac{1}{\log x}<\frac{1}{x}\sum\_{p\leq x}\log p<1+\frac{1}{2\log x},\qquad x\geq 41.$$


**Added.** See also the recent [paper of Platt-Trudgian](https://arxiv.org/abs/1407.1914) regarding the oscillation of the average in the middle around $1$.


