Geodesic
On complete rational trigonometric sums and integrals
Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 298-310.

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Asymptotical formulae as $m\to\infty$ for the number of solutions of the congruence system of a form $$ g_s(x_1)+\dots +g_s(x_k)\equiv g_s(x_1)+\dots +g_s(x_k)\pmod{p^m}, 1\leq s\leq n, $$ are found, where unknowns $x_1,\dots ,x_k,y_1,\dots ,y_k$ can take on values from the complete system of residues modulo $p^m,$ but degrees of polynomials $g_1(x),\dots ,g_n(x)$ do not exceed $n.$ Such polynomials $g_1(x),\dots ,g_n(x),$ for which these asymptotics hold as $2k>0,5n(n+1)+1,$ but as $2k\leq 0,5n(n+1)+1$ the given asymptotics have no place, were shew.Besides, for polynomials $g_1(x),\dots ,g_n(x)$ with real coefficients, moreover degrees of polynomials do not exceed $n,$ the asymptotic of a mean value of trigonometrical integrals of the form $$ \int\limits_0^1e^{2\pi if(x)}, f(x)=\alpha_1g_1(x)+\dots +\alpha_ng_n(x), $$ where the averaging is lead on all real parameters $\alpha_1,\dots ,\alpha_n,$ is found. This asymptotic holds for the power of the averaging $2k>0,5n(n+1)+1,$ but as $2k\leq 0,5n(n+1)+1$ it has no place.
Keywords: complete rational trigonometric sums, trigonometric integrals. 
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V. N. Chubarikov. On complete rational trigonometric sums and integrals. Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 298-310. http://geodesic.mathdoc.fr/item/CHEB_2018_19_3_a23/

[1] G. H. Hardy, S. Ramanujan, “Asymptotic formulae in combinatory analysis”, Proc. London math Soc. (2), 17 (1918), 75–115 | MR | Zbl

[2] G. H. Hardy, J. E. Littlewood, “A new solution of Waring problem”, Gött. Nachr., 1920, 33–54 | Zbl

[3] I. M. Vinogradov, “Sur un théorème génèral de Waring”, Mat. Sb., 31 (1924), 490–507 | Zbl

[4] I. M. Vinogradov, “On Waring's theorem”, Izv. AN SSSR, OFMN, 1928, no. 4, 393–400 | Zbl

[5] I. M. Vinogradov, The method of trigonometric sums in the theory of numbers, Nauka, M., 1980 | MR

[6] J. V. Linnik, Selected papers. The theory of numbers. The ergodic method and $L$-functions, Nauka, L., 1979, 432 pp. | MR | Zbl

[7] J. V. Linnik, “New estimations of Weyl sums”, Dokl. AN SSSR, 37:7 (1942), 201–203

[8] Hua Loo-Keng, Selected Papers, Springer-Verlag, N.-Y.–Heidelberg–Berlin, 1983, 888 pp. | MR | Zbl

[9] A. A. Karatsuba, N. M. Korobov, “On the mean-value theorem”, Dokl. AN SSSR, 149:2 (1963), 245–248 | Zbl

[10] A. A. Karatsuba, “Mean-value theorems and complete trigonometric sums”, Izv. AN SSSR, ser. math., 30 (1966), 183–206 | Zbl

[11] G. I. Arkhipov, Selected Papers, Publ. House of the Orjol State University, Orjol, 2013, 464 pp.

[12] G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, The theory of multiple trigonometric sums, Nauka, M., 1987 | MR

[13] G. I. Arkhipov, V. N. Chubarikov, A. A. Karatsuba, Trigonometric Sums in Number Theory and Analysis, de Gruyter Expositions in Mathematics, 39, Walter de Gruyter, Berlin–New York, 2004 | MR | Zbl

[14] V. N. Chubarikov, “The multiple complete rational arithmetical sums of polynomial values”, Dokl. RAN, 478:1 (2018), 22–24 | Zbl

[15] L. G. Arkhipova, V. N. Chubarikov, “The exponent of convergence of the singular series of a multivariate problem”, Bull. of Moscow State Univ. Ser. I, Math., mech., 2018, no. 5, 58–61