# The position of a prime between the two neighboring primes

Let $$p_0:=2\, \ldots\, p_{n-1}\,\,p_n\,\,p_{n+1}\,\ldots$$

be the increasing sequence of all primes. How often (three questions):

• $$\,p_n > \frac{p_{n-1}+p_{n+1}}2$$
• $$\,\sqrt{p_{n-1}\cdot p_{n+1}}\, <\, p_n\, \le\, \frac{p_{n-1}+p_{n+1}}2$$
• $$\, p_n < \sqrt{p_{n-1}\cdot p_{n+1}}$$

?

For the second question, assume that $$p_{n+1}-p_{n-1}<\sqrt{n}/10$$ (this is conjectured true for all sufficiently large $$n$$, but even just the prime number theorem says this is true of all but $$O(\sqrt{N}\log^2 N)$$ primes less than $$N$$).
Then $$\frac{p_{n-1}+p_{n+1}}{2}-\sqrt{p_{n-1}p_{n+1}}<0.5$$ so the question is asking how often $$p_{n-1},p_n,p_{n+1}$$ are in an arithmetic progression. The difference would have to be a multiple of 6 for all three numbers to be prime. You can get a good guess by summing over all possible multiples of six and using the Hardy-Littlewood conjecture.