# Distribution of the inbetween prime

Let $\ \mathbb J_n\,:=\,\{1\ \ldots\ n\}\$ be the initial interval of natural numbers, and $$2=p_0<p_1<\ldots$$ be the increasing sequence of all primes. Let

$$\forall_{n=1\ 2\ \ldots}\ \ d_n\, :=\, \frac{p_n-p_{n-1}}{p_{n+1}-p_{n-1}}$$

so that $\ p_n = (1-d_n)\cdot p_{n-1} + d_n\cdot p_{n+1}.\$Consider arbitrary reals: $$0<a<b<1$$

QUESTION

Can you prove or disprove or provide a correct limit or that there is none or a reference for equality:

$$\lim_{n=\infty}\ \frac 1n\cdot \left |\{k\in\mathbb J_n: a\le d_k\le b\}\right|\ =\ b-a$$

One may also consider the related questions about the frequency of the inequalities $\ p_n^2 > p_{n-1}\cdot p_{n+1}\$ or $\ p_{n+1}\cdot p_{n+2} > p_n\cdot p_{n+3},\$ or even some more precise questions too. (There is a slight bias toward the $\ >\$ inequality because it holds when the intermediate primes or their arithmetic mean are close to the arithmetic mean of the external pair of primes; but this wouldn't matter in the limit, or would it?)

• In your last limit line, shouldn't it be $a\le d_k \le b$, or have I misunderstood your claim? – Alan Jan 25 '16 at 19:42
• I dont understand. Do you mean $d_n$ or $d_k$? – Sylvain JULIEN Jan 25 '16 at 19:45
• I am so sorry--I fixed the typo, yes, there should be $d_k$, not $d_n$ at its last apperance of $d_?$. – Włodzimierz Holsztyński Jan 25 '16 at 19:49
• Perhaps you can get insights from the symetric density conjecture I consider in mathoverflow.net/questions/132973/… – Sylvain JULIEN Jan 25 '16 at 20:38
• I am struck by the thought that this is equivalent to the prime k-tuples conjecture. It probably isn't, but I could imagine some strong links between the two. Gerhard "Walk The Imaginary Bridge First" Paseman, 2016.01.25 – Gerhard Paseman Jan 26 '16 at 0:53