For a positive integer k, let d(k) be the number of divisors of k. So d(1)=1, d(p) =2 if p is a prime, d(6)=4, and d(12)=6.
What is the precise asymptotics of SUM_{k=1}^n 1/(kd(k))
Background:
1) This came up on the side in the polymath5 project.
2) There, Tim Gowers wrote: If nobody knows the answer, maybe that’s one for Mathoverflow, where I imagine a few minutes would be enough.
3) Asked: 14:17 Jerusalem time. (The first accurate answer: 17:44 Jerusalem time.)
4) Looking only at primes or only at integers with a typical number of divisors suggested a loglogn behavior, but looking at semiprimes indicates the sum is larger. I dont know how much larger.
5) I couldn't find an asnwer on the web. If there is an easy way searching for an answer that I missed this will be interesting too.
Follow up:
Great answers!! thanks. What about the sum
$\sum_{k=1}^n 1/(kd^2(k))$ ?
