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تسجيل مستخدم جديدApplying Discrete Fourier Transform to the Hardy-Littlewood Conjecture
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Jori Merikoski
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We study the asymptotic behaviour of the prime pair counting function $pi_{2k}(n)$ by the means of the discrete Fourier transform on $mathbb{Z}/ nmathbb{Z}$. 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 $mathbb{Z}/ nmathbb{Z}$ and on its subgroup $mathbb{Z}/ Qmathbb{Z},$ $Q , | , n.$
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In this note besides two abstra
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