Applying Discrete Fourier Transform to the Hardy-Littlewood Conjecture

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 $\mathbb{Z}/ Q\mathbb{Z},$ $Q \, | \, n.$