R\'epartition conjointe de trois nombres premiers et applications

TL;DR

This paper proves asymptotic results for the distribution of three primes, assuming a conjecture, and explores implications for primes in short intervals, advancing understanding in prime number theory.

Contribution

It provides new asymptotic formulas for three-prime distributions assuming a quantitative Hardy-Littlewood conjecture, extending previous work by Montgomery and Soundararajan.

Findings

Asymptotic results for the singular series of three primes.

Conditional formulas for joint prime distribution.

Potential applications to primes in short intervals.

Abstract

We prove asymptotic results for the singular series associated to the distribution of three primes. Assuming a quantitative version of Hardy and Littlewood's conjecture on prime 3-tuples, we deduce an asymptotic formula related to the joint distribution of three primes. This improves recent results of Kuperberg and completes results by Montgomery and Soundararajan. Following the Montgomery and Soundararajan approach, we derive conjectural applications to the distribution of primes in short intervals.