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2004-09-01, 15:44   #2
wblipp

"William"
May 2003
New Haven

93916 Posts
A Numerical Example with Known Results

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 x1/2 and the largest prime factor is less than x2/3.

Since the second largest prime can NEVER exceed x1/2, we know this is the same as the single condition that the largest prime factor is less than x2/3. And we know that this is rho(1.5). And we know that this is 1-ln(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 1-ln(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.