Complete Rational Arithmetic Sums / V. N. Chubarikov. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2016. № 1. P. 60-61 [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 43-44].

Let q>1 be an integer number, f(x)=a_nx^n+\ldots +a_1x+a_0 be a polynomial with integer coefficients, and (a_n,\ldots ,a_1,q)=1. The following estimate is valid:

\left|S \left(\frac{f(x)}{q}\right)\right|=\left|\sum_{x=1}^q\rho \left(\frac{f(x)}q\right)\right|\ll q^{1-1/n}, where \rho (t)=0,5-\{t\}.

Key words: "the saw-tooth" function, the Bernuolli's polynomials, the complete rational arithmetical sums.