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Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications
January 28, 2016 16:00–16:25, Dorodnitsyn Computing Centre, Department of Mechanics and Mathematics of Lomonosov Moscow State University., 119991, Moscow, Gubkina str., 8, Steklov Mathematical Institute, 9 floor, Conference hall

Contribution to the theory of hyperbolic zeta-functions of the lattices

N. M. Dobrovol'skii

Tula State Pedagogical University
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N. M. Dobrovol'skii

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Abstract: The talk is based on the results of common paper of N.M. Dobrovol'skii and N.N. Dobrovol'skii “On new results in the theory of hyperbolic zeta-function of lattices”, supported by grant No. 15-01-01540а of RFBR.

The hyperbolic zeta-function of tattices is defined in the right half-plane $\Re \alpha>1$ by series
$$ \zeta(\Lambda|\alpha) = \sum\limits_{\vec{x}\in \Lambda,\;\vec{x}\ne \vec{0}}(\overline{x}_{1}\ldots \overline{x}_{s})^{-\alpha}, $$
where $\overline{x}=\max{(1,|x|)}$. Obviously, in the case $s=1$ hyperbolic zeta-function of lattice is expressed in terms of Riemann zeta-function.

In [1], the following asymptotic formula for hyperbolic zeta-function of lattice$\Lambda(t,F)$ was derived:
$$ \zeta_{H}(\Lambda(t,F)|\alpha) = \frac{2(\det\Lambda(F))^{\alpha}}{R(s-1)!} ( \sum\limits_{(w)}|N(w)|^{- \alpha}) \frac{\ln^{s-1}{\det\Lambda(t,F)}}{(\det\Lambda(t,F))^{\alpha}} + O(\frac{\ln^{s-2}{\det\Lambda(t,F)}}{(\det\Lambda(t,F))^{\alpha}}). $$
Here $R$ denotes the regulator of the field $F$, and the sum over $(w)$ is the sum over all main ideals of the ring $\mathbf{Z}_{F}$.

Denote by $\zeta_{D_{0}}(\alpha|F)$ Dedekind zeta-function corresponding to main ideals of quadratic field $F$:
$$ \zeta_{D_{0}}(\alpha|F) = \sum\limits_{(\omega)}|N(\omega)|^{-\alpha}. $$
$$ \zeta_{D_{0}}(\alpha|F) = \sum\limits_{(\omega)}|N(\omega)|^{-\alpha}\ln{|N(\omega)|}. $$

Theorem. The following asymptotic relation holds:
$$ \zeta_{H}(\Lambda(t,F)|\alpha) = \frac{2(\det\Lambda)^{\alpha}\zeta_{D_{0}}(\alpha|F)}{R}\cdot \frac{\ln{\det\Lambda(t)}}{(\det\Lambda(t))^{\alpha}} - \frac{2(\det\Lambda)^{\alpha}}{R(\det\Lambda(t))^{\alpha}}  (\ln{\det{\Lambda}} + \zeta_{D_{0}}'(\alpha|F)) + \frac{2(\det{\Lambda})^{\alpha}\zeta_{D_{0}}(\alpha|F)}{(\det{\Lambda(t)})^{\alpha}}(\theta_{1}(\alpha) +  \frac{\theta_{2}(\alpha)}{\sinh{(\alpha R/2)}}), $$
where $|\theta_{1}(\alpha)|\le 1$ and $\varepsilon_{0}^{-\alpha/2}\le \theta_{2}(\alpha)\le \varepsilon_{0}^{\alpha/2}$, $\varepsilon_{0}$ denotes the fundamental unit of quadratic field $F$ and $R$ denotes the regulator of this field.

The proof of this assertion is contained in [2].

In the case of quadratic fields, the analyze of these results shows that the asymptotic formula for hyperbolic zeta-function of algebraic lattice can be improved.

For the case of quadratic fields, further researches should be directed to the study of Dedekind zeta-function of main ideals of quadratic fields and to its derivatives.

[1] N.M. Dobrovol'skii, V.S. Van'kova, S.L. Kozlova, Hyperbolic zeta-function of algebraic lattices. Preprint of VINITI 12.04.90 № 2327-B90.

[2] L.P. Dobrovol'skaya, M.N. Dobrovol'skii, N.M. Dobrovol'skii, N.N. Dobrovol'skii, Hyperbolic zeta-function of lattices of quadratic field. Chebysh. sb., 136:4 (2015), 100-149.

Language: English

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