Density of semiprimes on short intervals

Perhaps this is a trivial question, but I'm not an expert. Let

$$Q(m) = \bigl| \{ n : m\leq n \leq m + \log(m) \mbox{ and } n = p \cdot q\text{, where }p,q\text{ are prime} \} \bigr|$$

i.e., $Q(m)$ is the size of the set containing the numbers between $m$ and $m + \log(m)$ that are the product of two primes.

What is the value of the limit:

$$\lim_{m \rightarrow \infty} \frac{Q(m)}{\log(m)}$$

• This is not a trivial question, no worries. First, would you know how to solve that for $n=p$ primes, instead of $n=pq$? Do you think we can? – Julien Apr 15 '13 at 16:40
• @julien: for $n = p$ (n is prime) it should be zero. – Vor Apr 15 '13 at 21:59

The expected size of $Q(m)$ is $\log\log m$ and so $$\liminf_{m\to\infty}\frac{Q(m)}{\log m}=0.$$
I don't know that the $\limsup$ exists but it is probably 0.
• why the expected size is $\log \log m$? – Vor Apr 17 '13 at 7:26
• @Vor: There are $\sim n\log\log n/\log n$ semiprimes up to $n$ (a result of Landau), hence in each interval of length $\log n$ around $n$ there are on average $\sim\log\log n$ semiprimes. – Charles Apr 17 '13 at 15:16