  The investigation of the asymptotic behavior of trigonometric series near the
origin is a prominent topic in mathematical analysis. For trigonometric series
in one variable, this problem was exhaustively studied by various authors in a
series of publications dating back to the work of G. H. Hardy, 1928.
  Trigonometric series in several variables have got less attention. The aim of
the work is to partially fill this gap by finding the asymptotics of
trigonometric series in several variables with the terms, having a form of `one
minus the cosine' up to a decreasing power-like factor: \[
\sum_{z\in\mathbb{Z}^{d}\setminus\{0\}}\frac{1}{\|z\|^{d+\alpha}}\left(1-\cos\langle
z,\theta\rangle\right), \qquad \theta\in\mathbb{R}^{d}, \] where
$\langle\cdot,\cdot\rangle$ is the standard inner product and $\|\cdot\|$ is
the max-norm on $\mathbb{R}^{d}$.
  The approach developed in the paper is quite elementary and essentially
algebraic. It does not rely on the classic machinery of the asymptotic analysis
such as slowly varying functions, Tauberian theorems or the Abel transform.
However, in our case, it allows to obtain explicit expressions for the
asymptotics and to extend to the general case $d\ge 1$ classical results of G.
H. Hardy and other authors known for $d=1$.
