In the book by Elias M.Stein and Guido Weiss "Introduction to Fourier Analysis On Euclidean Spaces" one states in [page 268](https://books.google.com/books?id=xnIwDAAAQBAJ&pg=PA268) 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,
$$\underset{R\rightarrow\infty}\limsup \,\,|\underset{0<|m|<R}\sum |m|^{-(n/2)+1/2}e^{2\pi i m\cdot x}|=\infty$$

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 request 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."