Geodesic
Fractional Parts of the Function~$x/n$
Matematičeskie zametki, Tome 100 (2016) no. 5, pp. 744-756

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Asymptotic formulas for sums of values of some class of smooth functions of fractional parts of numbers of the form $x/n$, where the parameter $x$ increases unboundedly and the integer $n$ ranges over various subsets of the interval $[1,x]$, are obtained.
Keywords: fractional parts, asymptotic behavior, divisor problem, method of trigonometric sums. 
@article{MZM_2016_100_5_a9,
     author = {A. V. Shubin},
     title = {Fractional {Parts} of the {Function~}$x/n$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {744--756},
     publisher = {mathdoc},
     volume = {100},
     number = {5},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_100_5_a9/}
}
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A. V. Shubin. Fractional Parts of the Function~$x/n$. Matematičeskie zametki, Tome 100 (2016) no. 5, pp. 744-756. http://geodesic.mathdoc.fr/item/MZM_2016_100_5_a9/