Is there a prime between the double of two consecutive primes

Question

Can we say that there always exists a prime in between $2p_i$ and $2p_{i+1}$ $\forall i \in {N}$

It seems likely, and I thought that we could consider the inverse that, for a prime $p_i, \frac{p_i}{2}$ is in between two consecutive primes

Answer 1

There does not even exist any constant $C>0$ such that there is always a prime between $Cp_j$ and $Cp_{j+1}$. By the work of Zhang–Maynard–Tao, there exists a constant $D$ such that the number of prime pairs $p,p+D$ up to $x$ is $\gg x/\log^2 x$. On the other hand, an upper bound sieve shows that for each $0<A<CD$, the number of primes $p\le x$ for which $p, Cp+A, p+D$ are all prime is $\ll x/\log^3 x$. Therefore for most prime pairs $p,p+D$, there will be no prime between $Cp$ and $C(p+D)$.