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:
$$
\zeta_{H}(\Lambda(t,F)|\alpha)\,=\,\frac{2(\det\Lambda(F))^{\alpha}}{R(s-1)!}
\biggl(\,\sum\limits_{(w)}|N(w)|^{-\,\alpha}\biggr)\,\frac{\ln^{s-1}{\det\Lambda(t,F)}}{(\det\Lambda(t,F))^{\alpha}}\,+\,O\biggl(\frac{\ln^{s-2}{\det\Lambda(t,F)}}{(\det\Lambda(t,F))^{\alpha}}\biggr).
$$
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 $F$:
$$
\zeta_{D_{0}}(\alpha|F)\,=\,\sum\limits_{(\omega)}|N(\omega)|^{-\alpha}.
$$
Then
$$
\zeta_{D_{0}}(\alpha|F)\,=\,\sum\limits_{(\omega)}|N(\omega)|^{-\alpha}\ln{|N(\omega)|}.
$$
Theorem.The following asymptotic relation holds: $$
\zeta_{H}(\Lambda(t,F)|\alpha)\,=\,\frac{2(\det\Lambda)^{\alpha}\zeta_{D_{0}}(\alpha|F)}{R}\cdot
\frac{\ln{\det\Lambda(t)}}{(\det\Lambda(t))^{\alpha}}\,-\,\frac{2(\det\Lambda)^{\alpha}}{R(\det\Lambda(t))^{\alpha}}\,
\bigl(\ln{\det{\Lambda}}\,+\,\zeta_{D_{0}}'(\alpha|F))\,+\,\frac{2(\det{\Lambda})^{\alpha}\zeta_{D_{0}}(\alpha|F)}{(\det{\Lambda(t)})^{\alpha}}\biggl(\theta_{1}(\alpha)\,+\,
\frac{\theta_{2}(\alpha)}{\sinh{(\alpha R/2)}}\biggr),
$$ 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.
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