О современных проблемах теории гиперболической дзета-функции решёток

In the talk, we give a survey of unsolved problems of the theory of lattice hyperbolic zeta-function defined in the half-plane $\Re \alpha >1$ by the series
$$ \zeta(\Lambda|\alpha)\,=\,\sum\limits_{\vec{x}\in\Lambda,\;\vec{x}\ne \vec{0}}(\overline{x}_{1}\ldots \overline{x}_{s})^{-\,\alpha}, $$
where $\overline{x} = \max{(1,|x|)}$. In the case $s=1$, the hyperbolic zeta-function is expressed by Riemann zeta-function. Hovewer, the essentially new problems arise in a multidimensional case, and they have no analogues in the one-dimensional case.