Re: Number of factors
- --- In email@example.com,
Bernardo Boncompagni <RedGolpe@...> wrote:
> > log(log(N)) + B_1 - sum(prime q < p, 1/q) + o(1)For log(p) << log(N), the average of omega(n/p),
> Interestingly enough, if p>>1, sum=B_1+log(log(p))
> and we get log(log(N))-log(log(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:
David (per proxy Pafnuty Lvovich)