Quote:
Originally Posted by Bob Silverman
You can't just compute the probability that (say) the largestis less than x^k and the probability that the second largest is less than x^j and conclude that the probability that the largest is less than x^k
AND that the second largest is less than x^j is the product of the two...

Of course you can't.
And I didn't.
What I said is that we have two descriptions for the same set.
The Rho Description  This is the set of numbers whose largest prime factor is less than x
^{2/3}.
The Mu Description  This is the set of numbers that satisfy both
1. The largest prime factor is less than x
^{2/3}
AND
2. The second largest prime factor is less than x
^{1/2}.
Note we are talking about joint distributions, not joint densities, so we cannot make infereneces based on assuming the largest factor is near x
^{2/3}.
I claim these two sets are identical, and hence their probabilities should be the same. If you think I've made an error, please demonstrate a number that belongs to one set and not the other.
William