Distribution of lattice points on hyperboloids
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 168-177
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Consider the region $\Omega_0$ on the hyperboloid $1=b^2+ac$ defined by the conditions
$$
0\le a\le L_21,\quad 0\le\frac ba\le t_21.
$$
Let $r(n,\Omega_0)_pr$ be the number of integral points $(a,b,c)$ with $a=p$ (a prime) on the hyperboloid $n=b^2+ac$ ($n>0$ is an integer) such that $(a,b,c)/\sqrt n\in\Omega_0$. It is proved that for prime $P>P(\varepsilon)$, $\varepsilon>0$,
$$
(K-\Delta-\varepsilon)\frac P{\log P}\le r(P^2,\Omega_0)_{pr}\le(K+\Delta+\varepsilon)
\frac P{\log P},
$$
where
$$
K=2(t_2-t_1)(L_2-L_1),\quad\Delta=L^2_2\cdot\frac{2\pi}3.
$$
@article{ZNSL_2003_302_a10,
author = {O. M. Fomenko},
title = {Distribution of lattice points on hyperboloids},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {168--177},
publisher = {mathdoc},
volume = {302},
year = {2003},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_302_a10/}
}
O. M. Fomenko. Distribution of lattice points on hyperboloids. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 19, Tome 302 (2003), pp. 168-177. http://geodesic.mathdoc.fr/item/ZNSL_2003_302_a10/