Estimate for the product of primes less than n

Solution 1:

The reason the sum

$$ \sum_{k = 2}^{n} \frac{k}{\log k} $$

works as an estimate of the sum of all primes up to $n$ is because, roughly speaking, one on $\log N$ numbers of size around $N$ are prime. You are estimating

The sum of all primes of a given size

with the approximation

The sum of all numbers of that size, multiplied by (an estimate of) the proportion of them that are primes

(note that this method relies on the fact that the average of the primes of size around $N$ is roughly the same as the average of all numbers of size around $N$... specifically, that average is around $N$)

The analogous method for products is not dividing out by $\log i$: it is taking the $\log i$-th root: you meant to consider

$$ \prod_{k=2}^{n} k^{1 / \log k} $$

Of course, this isn't necessarily any easier to deal with. The thing to do is the one that is usually useful for products: take the logarithm. Consider

$$ \log \prod_{\substack{p=2 \\ p \text{ prime}}}^N p$$

Solution 2:

The summation formula at the top of Answer 1 isn't good necessarily percentage-wise.

For instance, if n = 14, the formula gives the estimate for the sum at about 50.6, but the sum of all of the primes from 2 to 13, inclusive, is actually 41.