[R.D. Silverman]

Originally Posted by CRGreathouse

Using (to avoid the extra variability from the small primes that brings) and searching an interval around 10²⁰ I find

This supports the intuition that Landau is better for small numbers of prime factors and Erdős-Kac better for large. In this case the crossover is surprisingly large (8 prime factors) but neither estimate is particularly accurate.

I would not expect it to be for composites that are only 20 digits.