# $\ \lim_{x\to ∞ }\frac{π(x)} { x^δ}$

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 $δ$ $∈ ( 0 , 1 )$

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

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.
• 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 – Souvik Dey Nov 2 '12 at 9:08
• 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. – Dan Brumleve Nov 2 '12 at 9:10