2
$\begingroup$

I am looking for sequences $\{a_n\} \subset \mathbb{N}$ with the following properties:

(1) $\displaystyle \# \{a_n \leq x \} = \frac{x}{\sqrt{\log x}} + O_\epsilon(x^{1/2 + \epsilon})$, and

(2) $\# \{a_n : a_n \equiv a \pmod{m}\} = \frac{x}{\phi(m)\sqrt{\log x}} + O_\epsilon(x^{1/2 + \epsilon})$ whenever $\gcd(a,m) = 1$.

In other words, $\{a_n\}$ looks like a thicker version of the primes.

Are there any such examples?

$\endgroup$
  • 2
    $\begingroup$ If the $a_n$ can have repeats, then you should just be able to repeat primes an appropriate number of times. If the $a_n$ are distinct, then there is no such sequence. Perhaps you could clarify the question. $\endgroup$ – Lucia Jan 19 '15 at 20:19
  • $\begingroup$ @Lucia What is the problem in the case of the $a_n$ distinct? Do you need all the modules $m$ to get the contradiction? $\endgroup$ – juan Jan 19 '15 at 20:41
  • 2
    $\begingroup$ @juan When the $a_n$ are distinct, then from the conditions it follows that very few of the $a_n$ can be multiples of a prime $p$ with $p\le x^{1/10}$ say. But then the number of integers up to $x$ composed of primes larger than $x^{1/10}$ is $O(x/\log x)$. $\endgroup$ – Lucia Jan 19 '15 at 21:53
  • $\begingroup$ That comment is satisfactory for my purposes, thank you $\endgroup$ – Stanley Yao Xiao Jan 20 '15 at 17:26

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.