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 $δ$ $∈ ( 0 , 1 )$
$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $
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prime-numbers
analytic-number-theory
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0What does $\pi\left(x\right)$ mean? – 2012-11-02
1 Answers
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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.
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0It 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. – 2012-11-02