Geodesic
On the distribution of fractional parts of polynomials
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 26, Tome 392 (2011), pp. 191-201 
O. M. Fomenko. On the distribution of fractional parts of polynomials. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 26, Tome 392 (2011), pp. 191-201. http://geodesic.mathdoc.fr/item/ZNSL_2011_392_a9/
@article{ZNSL_2011_392_a9,
     author = {O. M. Fomenko},
     title = {On the distribution of fractional parts of polynomials},
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     pages = {191--201},
     year = {2011},
     volume = {392},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_392_a9/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In the paper upper bounds for sums of the form $$ \sum_{0<n\le N}\psi\big(f(n)\big), $$ where $f(x)$ is a polynomial and $\psi(x)=x-[x]-1/2$, are obtained. The cases $$ f(x)=\frac1\alpha x^2+\beta x+\gamma $$ and $$ f(x)=\frac1\alpha x^3+\beta x^2+\gamma x+\delta $$ are considered, where $\alpha$ is a large positive number. Weyl's method and V. N. Popov's reasoning (Mat. Zametki, 18 (1975), 699–704) are used.

[1] I. M. Vinogradov, Izbrannye trudy, M., 1952

[2] T. Honda, “A few remarks on class numbers of imaginary quadratic number fields”, Osaka J. Math., 12 (1975), 19–21 | MR | Zbl

[3] I. M. Vinogradov, Osobye varianty metoda trigonometricheskikh summ, M., 1976 | MR

[4] A. O. Gelfond, Yu. V. Linnik, Elementarnye metody v analiticheskoi teorii chisel, M., 1962

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

[6] V. N. Popov, “O chisle tselykh tochek pod paraboloi”, Mat. zametki, 18:5 (1975), 699–704 | MR | Zbl

[7] H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, Providence, Rhode Island, 1994 | MR

[8] J. D. Vaaler, “Some extremal problems in Fourier analysis”, Bull. Amer. Math. Soc. (2), 12 (1985), 183–216 | DOI | MR | Zbl

[9] K. Prakhar, Raspredelenie prostykh chisel, M., 1967

[10] I. Miyawaki, “On the number of the lattice points in the area $0$, $0\le a^k/n$”, Osaka J. Math., 12 (1975), 647–671 | MR | Zbl