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

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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. 
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     author = {O. M. Fomenko},
     title = {Lattice points in the circle and the sphere},
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     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/

[1] M. N. Huxley, Area, lattice points, and exponential sums, Oxford, 1996 | MR

[2] D. R. Heath-Brown, “Lattice points in the sphere”, Number theory in progress, v. 2, de Gruyter, Berlin, 1999, 883–892 | MR | Zbl

[3] E. Landau, “Über die Gitterpunkte in einem Kreise”, Math. Zeit., 5 (1919), 319–320 | DOI | MR | Zbl

[4] G. H. Hardy, E. Landau, “The lattice points of a circle”, Proc. Royal Soc. A, 105 (1924), 244–258 | DOI | Zbl

[5] I. M. Vinogradov, Izbrannye trudy, M., 1952

[6] O. M. Fomenko, “O raspredelenii drobnykh chastei mnogochlenov ot dvukh peremennykh”, Zap. nauchn. semin. POMI, 404, 2012, 222–232 | MR

[7] R. A. Smith, “The average order of a class of arithmetic functions over arithmetic progressions with application to quadratic forms”, J. für die reine und angew. Math., 317 (1980), 74–87 | MR | Zbl

[8] E. Landau, Ausgewählte Abhandlungen zur Gitterpunktlehre, Berlin, 1962 | MR

[9] Z. I. Borevich, I. R. Shafarevich, Teoriya chisel, M., 1985 | MR

[10] R. A. Smith, “The circle problem in an arithmetic progression”, Canad. Math. Bull., 11 (1968), 175–184 | DOI | MR | Zbl

[11] M. Abramovits, I. Stigan, Spravochnik po spetsialnym funktsiyam, M., 1979

[12] O. M. Fomenko, “O dzeta-funktsii Epshteina, I”, Zap. nauchn. semin. POMI, 286, 2002, 169–178 ; “II”, Зап. научн. семин. ПОМИ, 371, 2009, 157–170 | MR | Zbl

[13] K. Ramachandra, A. Sankaranarayanan, “Hardy's theorem for zeta-functions of quadratic forms”, Proc. Indian Acad. Sci. (Math. Sci.), 106:3 (1996), 217–226 | DOI | MR | Zbl

[14] E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd edn., revised by D. R. Heath-Brown, New York, 1986 | MR

[15] S. A. Stepanov, Arifmetika algebraicheskikh krivykh, M., 1991 | MR

[16] I. M. Vinogradov, Osobye varianty metoda trigonometricheskikh summ, M., 1976 | MR

[17] V. Jarnik, “Über die Mittelwertsätze der Gitterpuntlehre. 5 Abhandlung”, C̆asopis pro pĕst. mat. a fys., 69 (1940), 148–174 | MR | Zbl

[18] Y.-K. Lau, “On the mean square formula of the error term for a class of arithmetical functions”, Monatsh. Math., 128 (1999), 111–129 | DOI | MR | Zbl

[19] A. Z. Valfish, Tselye tochki v mnogomernykh sharakh, Tbilisi, 1959 | MR

[20] J. L. Hafner, “On the average order of a class of arithmetical functions”, J. Number Theory, 15 (1982), 36–76 | DOI | MR | Zbl