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Old 2014-08-01, 19:09   #1
CRGreathouse's Avatar
Aug 2006

597910 Posts
Default Estimating the number of primes in a partially-factored number

I'm interested in using information about a number to estimate the likelihood of it having a certain number of prime factors.

If we had no information about the number except its size, there are two obvious approaches: the Erdős-Kac theorem plus continuity correction, or Landau's extension to the Prime Number Theorem:
\pi_k(n)\sim\frac{n}{\log n}\frac{(\log\log n)^{k-1}}{(k-1)!}

Hildebrand, Tenenbaum, and various collaborators have versions of this formula which works over a larger range, though all results are stated asymptotically so it's hard to tell over what ranges each would be best suited.

It's not obvious to me when I should use each of the various estimates. (I also wonder if there is an integral formulation of Landau's theorem, just like Li(x) is a better approximation to pi(x) than x/log x.)

Things get even more interesting if you add in partial factorization. First, it essentially takes care of the difference between distinct and indistinct factorizations, since beyond a fairly small point you don't see repeated prime factors of that size 'in the wild'. Second, it should be able to use the factorization information to refine the above estimates. Essentially you'd expect about log log x primes out of the first x, and you should be able to subtract this delta from the Erdős-Kac estimate directly. (It's not so clear how to modify Landau's formula.)

I ask these questions because in the range of numbers of interest (from those we can factor fairly easily to those we can check for probable primality fairly easily) the numbers are small enough that constant factors and o(1)'s matter.

And (pipe dream?) if the calculations are fairly routine we might ask Syd (or, for that matter, Alpertron) to include them in factordb (resp., his ECM page).
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