>

> Provided that p << N, I guess that the average value

> of omega(n/p) for the numbers n up to N with least

> prime divisor p is

>

> log(log(N)) + B_1 - sum(prime q < p, 1/q) + o(1)

>

> with B_1 =

> 0.261497212847642783755426838608695859051566648261199206192...

>

Interestingly enough, if p>>1, sum=B_1+log(log(p)) and we get

log(log(N))-log(log(p)).

The chance of N being prime is 1/log(N). Invoking Mertens' 3rd theorem, the

chance of a number sieved up to p being prime should be

log(p)/(log(N)*exp(-gamma)), where gamma is the Euler-Mascheroni constant.

If the log(N) in the number of divisors is distantly related to the chance

of N being prime, one may squeeze this result into our formula obtaining

log(log(N)*exp(-gamma)/log(p))=log(log(N))-log(log(p))-gamma. Not sure if

this last passage makes sense but at this point I guess

log(log(N))-log(log(p)) is a very reasonable answer to the problem. By the

way, N>p^2 means log(log(N))-log(log(p))>log(2) which looks reasonable

enough.

It's even very elegant if one writes it as log(lg(N)), where lg is the

logarithm to base p...

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