Geodesic
The distribution of lattice points on the four-dimensional sphere
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 16, Tome 263 (2000), pp. 226-236 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $r_l(n)$ be the number of representations of $n$ by a sum of $l$ squares of integers and let $0 be a constant. It is proved that if $(n,2)=1$, then $\sum_{-A\le w/\sqrt n\le A} r_3(n-w^2)=\mu_4(A)r_4(n)+O(n^{1487/2000}),\mu_4(A)>0$. Previously, the author obtained this asymptotics with a weaker error term $O(n^{3/4+\varepsilon})$. 
@article{ZNSL_2000_263_a14,
     author = {O. M. Fomenko},
     title = {The distribution of lattice points on the four-dimensional sphere},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {226--236},
     year = {2000},
     volume = {263},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_263_a14/}
}
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O. M. Fomenko. The distribution of lattice points on the four-dimensional sphere. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 16, Tome 263 (2000), pp. 226-236. http://geodesic.mathdoc.fr/item/ZNSL_2000_263_a14/