Real exponential sums over primes and prime gaps

Luan Alberto Ferreira

We prove that given $\hat{I}»\in \mathbb{R}$ such that $0<\hat{I}»<1$, then $\ddot{I}€(x+x^{\hat{I}}»)-\ddot{I}€(x)\sim \frac{x^{\hat{I}}»}{\log (x)}$. This solves a long-standing problem concerning th…