Real exponential sums over primes and prime gaps

We prove that given $\lambda \in \mathbb{R}$ such that $0<\lambda <1$, then $\pi (x+x^{\lambda })-\pi (x)\sim {\displaystyle \frac{x^{\lambda }}{\log (x)}}$. This solves a long-standing problem concerning the existence of primes in short intervals. In particular, we give a positive answer (for all sufficiently large number) to some old conjectures about prime numbers, such as Legendre’s conjecture about the existence of at least two primes between two consecutive squares.