Estimating the number of primes in a partially-factored number

[R.D. Silverman]

Originally Posted by CRGreathouse

Interesting.

I haven't worked through the proof yet. Do you recommend the original or a modern version (Granville-Soundarajan, etc.)?

I can't give a recommendation, I did not even know about the latter
result.

[CRGreathouse]

Originally Posted by R.D. Silverman

I can't give a recommendation, I did not even know about the latter
result.

I read it over last night -- Granville is a great expositor (in addition to being a first-rate mathematician). It doesn't look like their method easily extends to a correction term, since they're not finding the terms directly but rather the moments and then deriving the desired conclusion from a 'magical' theorem in statistics that says that if all the moments match the normal distribution it is normal. So it appears that 1 + o(1) is all you get.

But it struck me while reading the theorem that even more important than the correction term in variance is the correction term in mean, and it is purely elementary to improve the error term in the average order of from O(1) to M + O(1/log x), where the error term comes both from the fraction of numbers below x which are prime and from Mertens' theorem. I know this isn't the same as the normal order, and I doubt something so sharp could be proved at that level, but this should improve the practical performance.

[CRGreathouse]

Originally Posted by R.D. Silverman

I would try to get a 2nd term (i.e. main term plus error term with effective constant) for the
Erdos-Kac thm.

Look what I found:

Persi Diaconis, Frederick Mosteller, Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors—Including the square free case, Journal of Number Theory 9:2 (May 1977), pp. 187-202.

[R.D. Silverman]

Originally Posted by CRGreathouse

Look what I found:

Persi Diaconis, Frederick Mosteller, Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors—Including the square free case, Journal of Number Theory 9:2 (May 1977), pp. 187-202.

Excellent.

The authors surprise me. None of them, AFAIK, is a number theorist.

OTOH, I got to take statistics from Mosteller. He was a terrific
lecturer.

[CRGreathouse]

Originally Posted by R.D. Silverman

The authors surprise me. None of them, AFAIK, is a number theorist.

The only one I know is Diaconis who is a probabilist.