Quote:
Originally Posted by wblipp
Let's take a simple case where we know the answer. Consider the second largest prime is less then than x^{1/2} and the largest prime factor is less than x^{2/3}.
Since the second largest prime can NEVER exceed x^{1/2}, we know this is the same as the single condition that the largest prime factor is less than x^{2/3}. And we know that this is rho(1.5). And we know that this is 1ln(1.5) = 0.595.
For using the expression in the paper, we have alpha = 2 and beta = 1.333.
Over these ranges we know rho analytically as 1 or 1ln(t), so it's easy to evaluate both expressions.
The expression in the paper evaluates to 0.496. The expression in my guess evaluates to 0.595.

The probability that the second largest is less than x
^{1/B} is not
independent of the probability that the largest is less than x
^{1/A}.
The distribution of the second largest factor depends on the size of the
largest one... If the largest is near x
^{2/3} then the second
largest must be less than x
^{1/3}..... You have to consider the
joint pdf and they are not independent. Might this be the source of the
problem? You can't just compute the probability that (say) the largest
is 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...
Dan Bernstein has code that will do these computations. Perhaps you
might want to compare what his code yields with yours...
I am NOT saying that you are wrong. I could very well be in error.
But others have read this paper as well without finding what might be
this problem...