Geodesic
Elementary of the complete rational arithmetical sums
Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 450-459.

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In the paper the estimate of the complete rational arithmetical sum from a polynomial is found. It's is a correct on the power of a denominator with the estimate of the constant depending of the degree of a polynomial. Bibliography: 10 titles.
Keywords: the Gauss theorem of a multiplication for the Euler gamma-function, complete rational arithmetical sums, a functional equation on a complete system of residues by modulo of natural number, the Bernulli polynomials. 
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V. N. Chubarikov. Elementary of the complete rational arithmetical sums. Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 450-459. http://geodesic.mathdoc.fr/item/CHEB_2015_16_3_a21/

[1] Vinogradov I. M., The method of trigonometric sums in the theory of numbers, 2nd edition, Nauka, M., 1980, 144 pp. (in Russian)

[2] Chen Jing-run, “On Professor Hua's estimate on exponential sum”, Acta Sci. Sinica, 20:6 (1977), 711–719 | MR | Zbl

[3] Arkhipov G. I., Selected works, Orl. Gos. Univ., Orel, 2013, 464 pp. (in Russian)

[4] Arkhipov G. I., Chubarikov V. N., Karatsuba A. A., Trigonometric Sums in Number Theory and Analysis, De Gruyter expositions in mathematics, 39, Berlin–New York, 2004, 554 pp. | MR | Zbl

[5] Romanov N. P., Number theory and functional analysis, Collected papers, Tom. Univ., Tomsk, 2013, 478 pp. (in Russian)