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Is there any $\alpha>0$ for which there are known to exist two sequences of primes, $(p_i), (q_i)$ such that $$\alpha = \lim_{i\to\infty} \left(p_i/\ln p_i - q_i /\ln q_i\right)\ ?$$ The motivation here is to ask the simplest possible question about patterns in the primes after normalizing to "correct" for the prime number theorem.

More generally, one could do something like this. First look at the point set $P=\{ p/ \ln p\}$ where $p$ runs over all the primes. Then view this point set as an atomic measure on ${\Bbb R}$. Equip some nice set of measures on ${\Bbb R}$ with some appropriately weak topology and construct a dynamical flow bytaking the closure of all the left shifts of $P$.

Is anything non-trivial known about this flow?

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Some examples might be $q_i=2^i-1, p_i=2^i+i$ and $p_i=q_i + 2\lfloor C \log{ q_i} \rfloor$ though proving there are infinitely many such may be quite hard. 100 digits examples of the second kind are easy to find. If you allow $\alpha$ to be in certain range it might be doable. –  joro Jul 28 '12 at 11:08
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