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А.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
May 22, 2017 16:55, 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
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V. A. Bykovskii
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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}
$$ \sum\limits_{(a,b,c)\in K_{\mathbb{Z}}^{+}(d)}\varphi(\frac{b}{2c},\frac{\sqrt{|d|}}{2c}) = $$

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


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