On the difference between consecutive primes
  Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new results.   We show that the sum of squares of differences between consecutive primes $\sum_{p_n\le x}(p_{n+1}-p_n)^2$ is bounded by $x^{5/4+{\epsilon}}$ for $x$ sufficiently large and any fixed ${\epsilon}>0$. The proof relies on utilising various mean-value estimates for Dirichlet polynomials. 
J. Maynard
math.NT