Quote:
Originally Posted by Bob Silverman
I have just double checked the paper again and my notes. The expression
appears to be correct. Note that what we call alpha is actually the
reciprocal of what Knuth uses. Could this be a source of confusion?
Perhaps the confusion is mine. I am certainly not infallible. But Sam Wagstaff
checked the paper carefully as well...

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.