The Bateman-Horn Conjecture: Heuristics, History, and Applications

TL;DR

The paper explores the Bateman-Horn conjecture, a significant hypothesis about prime number distribution, discussing its implications, historical background, and various applications in number theory.

Contribution

It provides a comprehensive overview of the conjecture's origins, its implications for prime distribution, and how it connects to major results and open problems in mathematics.

Findings

Links the conjecture to the prime number theorem and Green-Tao theorem

Suggests the conjecture implies twin prime and Landau's conjectures

Highlights the conjecture's role in understanding prime distribution

Abstract

The Bateman-Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green-Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau's conjecture. We discuss the Bateman-Horn conjecture, its applications, and its origins.