Inverse of Smoothness Probability

[paul0]

The probability that a random number below X is B-smooth is given by u^-u, where u=ln(X)/ln(B). However, I would like the do the inverse, that is, given the smoothness probability and B, how do I solve for X?

I have a solution via Newton method. Is there a closed form?

[Batalov]

Originally Posted by paul0

The estimate of the probability that a random number below X is B-smooth is given by u^-u, where u=ln(X)/ln(B).

No need for Newton method, because it will immediately exceed the precision of the estimate that you started with. There are better estimates (and then see Dickman's and/or Buchstab's).

[paul0]

Originally Posted by Batalov

No need for Newton method, because it will immediately exceed the precision of the estimate that you started with. There are better estimates (and then see Dickman's and/or Buchstab's).

I think you misunderstood. Instead of solving the probability of smoothness, I want to solve for X given B and the probability.

[Dubslow]

Originally Posted by paul0

I think you misunderstood. Instead of solving the probability of smoothness, I want to solve for X given B and the probability.

He's saying that going in reverse on that equation (p = some_function_of(X, B), as you're trying to do, isn't very useful because it's not an equation -- only an approximation, and not a very good one. Better to write p ~ some_function_of(X, B), and then note that the "~" is vague enough that undoing "some_function_of" isn't worth the effort.

The links he provides are sources for p ≈ some_better_function_of(X, B). Find those better functions before trying to invert them.

[paul0]

Originally Posted by Dubslow

He's saying that going in reverse on that equation (p = some_function_of(X, B), as you're trying to do, isn't very useful because it's not an equation -- only an approximation, and not a very good one. Better to write p ~ some_function_of(X, B), and then note that the "~" is vague enough that undoing "some_function_of" isn't worth the effort.

The links he provides are sources for p ≈ some_better_function_of(X, B). Find those better functions before trying to invert them.

I stand corrected. Thanks for clarifying :)

[Batalov]

Incidentally, I have just heard from much more knowledgeable people, and I will simply quote:

Originally Posted by Robert

Note that the question (u^-u) can be answered using the Lambert W function. But, of course, one still must compute the answer numerically via series/Pade approximant, etc.
Note that the W function is not real-analytic.

Bob

[Batalov]

And for the Inverse of Smoothness Probability question, you want a reasonable numerical estimate the inverse of Dickman function.
So you want to take Newton method on the inverse of Dickman ρ - because if you know the derivative of Dickman ρ by definition, then you know the derivative of the inverse of Dickman ρ.