No Arabic abstract
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.
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 $mathbb{Z}/ Qmathbb{Z},$ $Q , | , n.$
In this note besides two abstra
This paper is devoted to a new family of reverse Hardy-Littlewood-Sobolev inequalities which involve a power law kernel with positive exponent. We investigate the range of the admissible parameters and characterize the optimal functions. A striking open question is the possibility of concentration which is analyzed and related with nonlinear diffusion equations involving mean field drifts.
We consider Littlewood-Paley functions associated with non-isotropic dilations. We prove that they can be used to characterize the parabolic Hardy spaces of Calder{o}n-Torchinsky.
There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $lambda=n-alpha$ (that is for the case of $alpha>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.