Geodesic
On the distribution of integral points on cones
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 25, Tome 383 (2010), pp. 193-203 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $r_k(n)$ denote the number of representations of a positive integer $n$ as the sum of $k$ squares. We prove that $$ \sum_{n\le x}r^2_3(n)=Cx^2+O\Big(x^\frac32\big(\log x\big)^\frac72\Big), $$ where $C>0$ is a certain constant, and that $$ \sum_{n\le x}r^2_4(n)=32\zeta(3)x^3+O\Big(x^2\big(\log x\big)^\frac53\Big). $$ Bibl. 14 titles. 
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     author = {O. M. Fomenko},
     title = {On the distribution of integral points on cones},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_383_a12/}
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O. M. Fomenko. On the distribution of integral points on cones. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 25, Tome 383 (2010), pp. 193-203. http://geodesic.mathdoc.fr/item/ZNSL_2010_383_a12/

[1] O. M. Fomenko, “Raspredelenie tselykh tochek na nekotorykh poverkhnostyakh vtorogo poryadka”, Zap. nauchn. semin. POMI, 212, 1994, 164–195 | MR | Zbl

[2] M. Kühleitner, W. G. Nowak, “The averange number of solutions of the Diophantine equation $U^2+V^2=W^3$ and related arithmetic functions”, Acta Math. Hungar., 104:3 (2004), 225–240 | DOI | MR | Zbl

[3] A. Schinzel, “On an analytic problem considered by Sierpiński and Ramanujan”, New trends in probability and statistics (Palanga, 1991), v. 2, VSP, Utrecht, 1992, 165–171 | MR

[4] W. Müller, “The mean square of Dirichlet series associated with automorphic forms”, Monatsh. Math., 113:2 (1992), 121–159 | DOI | MR | Zbl

[5] W. Müller, “The Rankin–Selberg method for non-holomorphic automorphic forms”, J. Number Theory, 51:1 (1995), 48–86 | DOI | MR | Zbl

[6] Khua Lo-gen, Metod trigonometricheskikh summ i ego primeneniya v teorii chisel, M., 1964

[7] P. T. Bateman, “On the representations of a number as the sum of three squares”, Trans. Amer. Math. Soc., 71:1 (1951), 70–101 | DOI | MR | Zbl

[8] O. Saparniyazov, A. S. Fainleib, “Dispersiya summ deistvitelnykh kharakterov i momenty $L(1,\chi)$”, Izvestiya AN Uzb.SSR. Seriya fiz.-mat. nauk, 1975, no. 6, 24–29 | Zbl

[9] M. Jutila, “On character sums and class numbers”, J. Number Theory, 5:3 (1973), 203–214 | DOI | MR | Zbl

[10] J. L. Hafner, “On the representation of the summatory functions of a class of arithmetical functions”, Lect. Notes Math., 899, 1981, 148–165 | DOI | MR | Zbl

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

[12] S. Ramanujan, “Some formulae in the analytic theory of numbers”, Messenger Math., 45 (1916), 81–84

[13] R. A. Smith, “An error term of Ramanujan”, J. Number Theory, 2:1 (1970), 91–96 | DOI | MR | Zbl

[14] A. Z. Valfish, Tselye tochki v mnogomernykh sharakh, Tbilisi, 1960 | MR