Geodesic
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 Cet article a éte moissonné depuis la source Math-Net.Ru

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Consider the region $\Omega_0$ on the hyperboloid $1=b^2+ac$ defined by the conditions $$ 0<L_1\le a\le L_2<1,\quad 0<t_1\le\frac ba\le t_2<1. $$ 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 $$ K=2(t_2-t_1)(L_2-L_1),\quad\Delta=L^2_2\cdot\frac{2\pi}3. $$ 
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     author = {O. M. Fomenko},
     title = {Distribution of lattice points on hyperboloids},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_302_a10/}
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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/

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