Geodesic
Lattice points in the circle and the sphere
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 28, Tome 418 (2013), pp. 198-220

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Let $P(x)$ and $P_3(x)$ be the error terms in the Gaussian circle problem and the sphere problem, respectively. We investigate the asymptotic behavior of the sums $$ \sum_{\substack{k\le x\\k\equiv0\!\!\!\pmod p}}P(k),\quad\sum_{\substack{k\le x\\k\equiv0\!\!\!\pmod p}}P_3(k). $$ Here $p\ge2$ is a prime number. 
@article{ZNSL_2013_418_a13,
     author = {O. M. Fomenko},
     title = {Lattice points in the circle and the sphere},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {198--220},
     publisher = {mathdoc},
     volume = {418},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_418_a13/}
}
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O. M. Fomenko. Lattice points in the circle and the sphere. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 28, Tome 418 (2013), pp. 198-220. http://geodesic.mathdoc.fr/item/ZNSL_2013_418_a13/