Applying Discrete Fourier Transform to the Hardy-Littlewood Conjecture

TL;DR

This paper introduces a novel discrete Fourier transform approach on modular integers to analyze prime pair distributions, providing insights aligned with the Hardy-Littlewood Conjecture and offering advantages over traditional circle methods.

Contribution

It develops a discrete Fourier transform method on or analyzing prime pairs, offering a new perspective compared to classical circle methods.

Findings

Recovered the main term for prime pair counts predicted by Hardy-Littlewood.

Demonstrated advantages of the discrete Fourier approach over Fourier series.

Provided a framework connecting Fourier transforms on or subgroup analysis.

Abstract

We study the asymptotic behaviour of the prime pair counting function $\pi_{2k}(n)$ by the means of the discrete Fourier transform on $\mathbb{Z}/ n\mathbb{Z}$. The method we develop can be viewed as a discrete analog of the Hardy-Littlewood circle method. We discuss some advantages this has over the Fourier series on $\mathbb{R} /\mathbb{Z}$, which is used in the circle method. We show how to recover the main term for $\pi_{2k}(n)$ predicted by the Hardy-Littlewood Conjecture from the discrete Fourier series. The arguments rely on interplay of Fourier transforms on $\mathbb{Z}/ n\mathbb{Z}$ and on its subgroup $\mathbb{Z}/ Q\mathbb{Z},$ $Q \, | \, n.$