I have a sequence $ a_{n}$, and the set

$ A=\{ m \le x | m \in \{a_{n}\}_{n=0}^{\infty} \} $ satisfying

$ \#A= x^\alpha+O(x^{\alpha-\varepsilon}). $

Also I have

$ \#\{ m \le x : p|m , m \in \{a_{n}\}_{n=0}^{\infty} \}= \frac{x^\alpha}{p}+O(x^{\alpha-\varepsilon}). $ for all p prime (of course those $p \ll x^{\varepsilon}$).

Is there any sieve(or maybe direct) theoritical approach from which I can conclude that R-smooth numbers in the above set has positive lower density(maybe asymptotic)?

Well I did try Buchstab's identity together with usual induction argument. I got no result.



1 Answer 1


Let $A$ be the set of integers $n$ which are divisible by a prime number $p>n^\theta$, $\frac{1}{2}<\theta<1$. Then for a prime number $q<x^{1/3}$ we have \begin{eqnarray*} \#\{n\leq x, n\in A, q|n\} & = & \underset{p\neq q}{\sum_{p\leq x}}\underset{q|m}{\sum_{m\leq\min(p^{(1-\theta)/\theta}, x/p)}}1 + \mathcal{O}(q^2)\\ & = & \sum_{p\leq x}\left\lfloor\frac{\min(p^{(1-\theta)/\theta}, x/p)}{q}\right\rfloor+ \mathcal{O}(q^2)\\ & = & \sum_{x^\theta\leq p\leq x}\frac{x}{pq} + \mathcal{O}\left(\frac{x}{\log x}\right). \end{eqnarray*} Hence $A$ is well distributed in residue classes, but does not contain smooth numbers.


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.