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Let $ \pi(x)$ denote the prime counting function.Then what is $\lim\limits_{n\to\infty}sup\space\Big (\frac {π(2n+1)}{π(2n)}\Big)^n $ ?

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    @GerryMyerson: Oh , right , I did not notice.2012-11-09

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$\pi(x)$ is unbounded above, and $\pi(2n+1)=\pi(2n)$ except on a subsequence of density 0. So $ \liminf_{n\to\infty}n\log\frac{\pi(2n+1)}{\pi(2n)}=\liminf_{n\to\infty}n\log1=0. $

On the other side it suffices to look at the subsequence of primes: $ \limsup_{n\to\infty}n\log\frac{\pi(2n+1)}{\pi(2n)}=\limsup_{n\to\infty}\frac{p_{n+1}-1}{2}\log\frac{n+1}{n}=\limsup_{n\to\infty}\frac{p_{n+1}-1}{2}\left(\frac{1}{n}+O\left(\frac{1}{n^2}\right)\right)=+\infty $