Abstract:
Let $q>1$ be an integer, $f(x)=a_nx^n+\ldots +a_1x+a_0$ be a polynomial with the integer coefficients, and $(a_n,\ldots ,a_1,q)=1.$ Then is valid the estimation $$\left|S\left(\frac{f(x)}{q}\right)\right|=\left|\sum_{x=1}^q\rho\left(\frac{f(x)}q\right)\right|\ll q^{1-1/n}, $$ where $\rho(t)=0,5-\{t\}.$
Key words:“the saw-tooth” function, the Bernuolli's polynomials, the complete rational arithmetical sums.
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