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Register a new userApplying Discrete Fourier Transform to the Hardy-Littlewood Conjecture
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Authors
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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We study the prime pair counting functions $pi_{2k}(x),$ and their averages over $2k.$ We show that good results can be achieved with relatively little effort by considering averages. We prove an asymptotic relation for longer averages of $pi_{2k}(x)$ over $2k leq x^theta,$ $theta > 7/12,$ and give an almost sharp lower bound for fairly short averages over $k leq C log x,$ $C >1/2.$ We generalize the ideas to other related problems.
The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the DFT. The transition matrix from the standard basis to the canonical basis defines a novel transform which we call the discrete oscillator transform (DOT for short). Finally, we describe a fast algorithm for computing the discrete oscillator transform in certain cases.
In this work, we present two parallel algorithms for the large-scale discrete Fourier transform (DFT) on Tensor Processing Unit (TPU) clusters. The two parallel algorithms are associated with two formulations of DFT: one is based on the Kronecker product, to be specific, dense matrix multiplications between the input data and the Vandermonde matrix, denoted as KDFT in this work; the other is based on the famous Cooley-Tukey algorithm and phase adjustment, denoted as FFT in this work. Both KDFT and FFT formulations take full advantage of TPUs strength in matrix multiplications. The KDFT formulation allows direct use of nonuniform inputs without additional step. In the two parallel algorithms, the same strategy of data decomposition is applied to the input data. Through the data decomposition, the dense matrix multiplications in KDFT and FFT are kept local within TPU cores, which can be performed completely in parallel. The communication among TPU cores is achieved through the one-shuffle scheme in both parallel algorithms, with which sending and receiving data takes place simultaneously between two neighboring cores and along the same direction on the interconnect network. The one-shuffle scheme is designed for the interconnect topology of TPU clusters, minimizing the time required by the communication among TPU cores. Both KDFT and FFT are implemented in TensorFlow. The three-dimensional complex DFT is performed on an example of dimension $8192 times 8192 times 8192$ with a full TPU Pod: the run time of KDFT is 12.66 seconds and that of FFT is 8.3 seconds. Scaling analysis is provided to demonstrate the high parallel efficiency of the two DFT implementations on TPUs.
This paper deals with the inequalities devoted to the comparison between the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley (Pitt) inequalities in the noncommutative context. We establish Hormanders $L^p$-$L^q$ Fourier multiplier theorem on compact hypergroups for $1<p leq 2 leq q<infty$ as an application of Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conjugacy classes of compact Lie groups and for a class of countable compact hypergroups.
In this note besides two abstra
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