There is a function for approximating the number of primes below a certain integer
What would be the function for determining the chances of rolling a prime number when rolling n number of d-sided dice?
If I throw four twenty-sided dice, the chances that the values will add up to a prime number can be precisely answered, but what is the equation using only the $\frac{x}{log(x)}$ approximation of the prime-counting function?
There really isn't a clean approximation. The exact formula would be $$\sum_{i=1}^{nd}P(\text{X = i})I_{\text{primes}}(i)$$ where $X$ follows an Irwin-Hall distribution (approximately normal) and $I_{\text{primes}}(i)$ is the indicator function denoting whether $i$ is prime. You cannot apply the $x/\log(x)$ insight because it would require the distribution of $X$ is uniform, which is clearly violated.