4
$\begingroup$

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:

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

$\endgroup$
2
  • 1
    $\begingroup$ 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? $\endgroup$ – Julien Apr 15 '13 at 16:40
  • $\begingroup$ @julien: for $n = p$ (n is prime) it should be zero. $\endgroup$ – Vor Apr 15 '13 at 21:59
4
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ why the expected size is $\log \log m$? $\endgroup$ – Vor Apr 17 '13 at 7:26
  • 1
    $\begingroup$ @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. $\endgroup$ – Charles Apr 17 '13 at 15:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.