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Conference in memory of A. A. Karatsuba on number theory and applications, 2015
January 30, 2015 12:30, Moscow, Steklov Mathematical Institute of the Russian Academy of Sciences
 


О современных проблемах теории гиперболической дзета-функции решёток

N. M. Dobrovol'skii

Tula State Pedagogical University
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N. M. Dobrovol'skii


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Abstract: 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.

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