Real exponential sums over primes and prime gaps

TL;DR

The paper addresses whether primes can be detected in extremely short intervals of the form $(x, x+x^\lambda]$ for $0<\lambda<1$. It develops an adaptation of Newman’s proof of the prime number theorem using a carefully crafted exponential-weighted prime-sum $W(x)$ and a Laplace-transform framework, showing that $W(x) \sim \exp(cx^{1-\lambda})$ leads to the target asymptotic $\pi(x+x^\lambda) - \pi(x) \sim \frac{x^\lambda}{\log x}$. The main contributions include proving the short-interval asymptotic, deriving a weighted identity that drives the result, and outlining a general method that connects to classic conjectures (e.g., Legendre) in the large-$x$ regime. The work provides a conceptual framework for primes in short intervals and suggests broad potential extensions and limitations, with implications for longstanding conjectures in number theory.

Abstract

We prove that given $λ\in \mathbb{R}$ such that $0 < λ< 1$, then $π(x + x^λ) - π(x) \sim \displaystyle \frac{x^λ}{\log(x)}$. 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.

Theorems & Definitions (60)

• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Corollary 2.5
• proof
• Theorem 2.6: Rosser-Schoenfeld