In the book by Elias M.Stein and Guido Weiss "Introduction to Fourier Analysis on Euclidean Spaces" one states in page 268 the following theorem:

Theorem 1:The trigonometric series $$\underset{m\in \mathbb{Z}^n, m\neq 0}\sum |m|^{-(n/2)+1/2}e^{2\pi i m\cdot x}$$ diverges almost everywhere. More particularly, $$\underset{R\rightarrow\infty}\limsup \,\,\bigg|\underset{0<|m|<R}\sum |m|^{-(n/2)+1/2}e^{2\pi i m\cdot x}\bigg|=\infty$$

We have noted in the previous theorem for $m=(m_1,\ldots,m_n)\in\mathbb{Z}^n$ and $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$:
$$|m|^{2}=m_{1}^{2}+\cdots+m_{n}^{2}$$ $$m\cdot x=m_{1}x_{1}+\cdots+m_{n}x_{n}.$$ My question is: For a real number $\alpha>0$ consider the trigonometric series $$\underset{m\in \mathbb{Z}^{n}, m\neq 0}\sum \frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}}.$$ For what value of $\alpha$ the previous series is convergent?

The convergence mode we adopt is the following: The previous series is convergent for $\alpha$ if $$\underset{R\rightarrow\infty}\lim {\underset{0<|m|\leq R}\sum \frac{e^{2\pi i m\cdot x}}{|m|^{\alpha}} }$$ exits.
The question is already well elucidated for $n=1$. For $n>1$ the previous theorem asserts that there is divergence for $\alpha=\frac{n}{2}-\frac{1}{2}$.
For other kind of convergence mode these question is treated by Stephen Wainger in "Special Trigonometric Series In k-Dimensions."