2
$\begingroup$

Let $\ π(x)$ denote the prime counting function , i.e. the number of primes not exceeding $x$
Then does $$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $$ exist for all real $δ$ $∈ ( 0 , 1 )$

$\endgroup$
  • $\begingroup$ What does $\pi\left(x\right)$ mean? $\endgroup$ – yiyi Nov 2 '12 at 9:09
2
$\begingroup$

No, because the prime number theorem states that $\pi(x) \approx \frac{x}{\log{x}}$. Since $\log{x} \lt x^\delta$ for any $\delta \gt 0$ and sufficiently large $x$, the limit diverges.

$\endgroup$
  • 1
    $\begingroup$ I know the asymptotic nature of $π(x)$ but can not really seem to get the proof for divergence of the said limit , can you please give it more formally $\endgroup$ – Souvik Dey Nov 2 '12 at 9:08
  • $\begingroup$ It follows algebraically from PNT and $\log{x} \lt x^\delta$ for all sufficiently large $x$ and $\delta \gt 0$, which I won't show, but you should be able to find references for the fact on this site. $\endgroup$ – Dan Brumleve Nov 2 '12 at 9:10

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.