Real exponential sums over primes and prime gaps

TL;DR

This paper proves that primes are densely distributed in short intervals of the form [x, x + x^λ] for 0 < λ < 1, confirming long-standing conjectures like Legendre's.

Contribution

It establishes the asymptotic density of primes in short intervals for all λ between 0 and 1, solving a major open problem in number theory.

Findings

Primes are asymptotically evenly distributed in short intervals.

Confirms Legendre's conjecture for sufficiently large numbers.

Provides new insights into prime gaps and distribution patterns.

Abstract

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.