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А.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
May 22, 2017 16:55–17:25, Moscow, Steklov Mathematical Institute

The distribution of lattice points on the hyperboloid

V. A. Bykovskii

Khabarovsk Division of the Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences
 Video records: MP4 717.0 Mb MP4 182.0 Mb

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Abstract: Let $d$ be an integer. Denote by
$$K_{\mathbb{Z}}(d) = \{(a,b,c)\in \mathbb{Z}^{3} \: b^{2}-4ac = d\}$$
the set of lattice points lying on the hyperboloid
$$\{(x_{1},x_{2},x_{3})\in \mathbb{R}^{3} : x_{2}^{2}-4x_{1}x_{2} = d\},$$
which is hyperbolic in the case $d<0$ and elliptic in the case $d>0$. Also, we denote
$$K_{\mathbb{Z}}^{+}(d) = \{(a,b,c)\in K_{\mathbb{Z}}(d) \: c>0\}.$$
The set $K_{\mathbb{Z}}(d)$ is non-empty if and only if $d\equiv 0,1 \pmod 4$. Such numbers $d\ne 0$ are called as discriminants because the elements of the set $K_{\mathbb{Z}}(d)$ parametrize the binary quadratic forms $Q(u,v) = au^{2}+buv+cv^{2}$ of discriminant $d$ with integers coefficients. Next, let $\delta_{q}(m) = 1$ if $m\equiv 0 \pmod q$, and $\delta_{q}(m) = 0$ otherwise. Then Dirichlet series
$$\sum\limits_{c=1}^{+\infty}( \sum\limits_{b \pmod{2c}}\delta_{4c}(b^{2}-d))\frac{1}{c^{s}} = \frac{\zeta(s)}{\zeta(2s)} G_{d}(s)\quad (d\ne n^{2})$$
converges absolutely in the half -plane $\Re s > 1$ and determines the entire function $G_{d}(s)$. The following theorem holds true:
Theorem. Suppose that $d\ne n^{2}$, $d\equiv 0,1 \pmod 4$, and let $\varphi(x,y)$ be any smooth complex-valued function over $\mathbb{R}\times (0,+\infty)$ with a compact support. Then, for any fixed $\varepsilon > 0$ we have}
$$\sum\limits_{(a,b,c)\in K_{\mathbb{Z}}^{+}(d)}\varphi(\frac{b}{2c},\frac{\sqrt{|d|}}{2c}) =$$

$$= \frac{3}{\pi^{2\mathstrut}} \sqrt{|d|}G_{d}(1)\int_{-\infty}^{+\infty}\int_{0}^{+\infty}\varphi(x,y) \frac{dx dy}{y^{2\mathstrut}} + O_{\varphi,\varepsilon}(|d|^{1/2-1/12+\varepsilon}).$$

Language: English

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