--- In

primenumbers@yahoogroups.com,

Bernardo Boncompagni <RedGolpe@...> wrote:

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

>

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

> and we get log(log(N))-log(log(p)).

For log(p) << log(N), the average of omega(n/p),

with n running from p to N and p the least

prime divisor of n, is (I claim) asymptotic to the

sum of 1/q, with prime q running from q = p up to

the greatest prime q <= N/p.

All I did was to use Mertens at the upper limit.

For log(2) << log(p) << log(N), one may also use

Mertens at the lower limit to get log(log(N)/log(p)),

as Bernardo has observed. However one should not push

this up to p = O(sqrt(N)), since there we may run into

a problem that worried Chebyshev:

http://tech.groups.yahoo.com/group/primenumbers/message/21565
David (per proxy Pafnuty Lvovich)