On some contact points with the multiple Fourier series and lattice point problems with weights

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For the multiple Fourier series of periodization of radial functions on d-dimensional Euclidean space, we studied the behavior of the spherical partial sum. We showed the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third phenomenon for the multiple Fourier series. The third phenomenon is closely related to the lattice point problems, which is classical theme of analytic number theory. We also proved that the convergence problems on the Fourier series is equivalent to the lattice point problems in a sense.