I'm trying to estimate the product
$$\prod\_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$
where $p,q,r,s$ are primes.


This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea is that there are 4! congruence classes mod $(pqrs)^2$ such that $n,n+1,n+2,n+3$ are each congruent to 0 mod the square of one of the primes. A number is not the first of four such numbers exactly when it is not in any of these congruence classes for any quadruple of distinct primes.


I feel that there should be some way to accelerate the calculation, possibly using the prime zeta function. At the least, this is useful for the sum approximation which is far easier to calculate with dynamic programming (making it essentially linear rather than quartic, if you can spare $O(n)$ memory). Unfortunately there are enough congruence classes that the cancellation is an important part of the problem, so the sum approximation is poor.


Any suggestions for the calculation, or different approaches to the problem, would be appreciated.


[1] <http://oeis.org/classic/A070284>



Product over the primes
Modulo errors got unconditionally slightly smaller upper bound than your conjecture.
`prodeulerrat` from [Cohen's pari script](http://pari.math.u-bordeaux.fr/Scripts/) can compute $\prod\_{\text{p prime}}{1-1/O(x^2)}$.


For fixed $p < q < r < L$ define $g(p,q,r)=\prod\_{\text{s prime}}{1-24/(pqrs)^2}$


$g$ is efficiently computable by the script and contains the contribution
of $s > L$ times extra factors $ < L$. Compute the extra factors the slow way
for each triple to get the exact product for $p < q < r < L < s$ for fixed $p,q,r$. Upper bound (since all terms are $<1$) for your product is the product over the exact products times the product $< L$.
.


For $L=200$ got $0.9974805703680839724432166987$


