Geodesic
Estimation of the mean value of the remainder term in~the~asymptotic~formula for the sum of values of~an~arithmetical~function on a Beatty sequence
Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 523-528

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The paper is concerned with the estimation of average values of $\Delta(\alpha,N)=\Delta(\alpha,0,N)$ and $\Delta(\alpha,\beta,N)$ with respect to $\alpha>1$ and $0\beta\alpha$ respectively, where $\Delta(\alpha,\beta,N)$ denotes the remainder term in the formula of the form $$\sum_{n\leq N}f([\alpha n+\beta])=\frac{1}{\alpha}\sum_{m\leq \alpha N+\beta}f(m)+\Delta(\alpha,\beta,N),$$ for an arbitrary number-theoretical fuction $f(n)$.
Keywords: Beatty sequences, integer sequence, mean value of a number-theoretic function. 
@article{CHEB_2018_19_2_a35,
     author = {A. V. Begunts and D. V. Goryashin},
     title = {Estimation of the mean value of the remainder term in~the~asymptotic~formula for the sum of values of~an~arithmetical~function on a {Beatty} sequence},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {523--528},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2018_19_2_a35/}
}
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A. V. Begunts; D. V. Goryashin. Estimation of the mean value of the remainder term in~the~asymptotic~formula for the sum of values of~an~arithmetical~function on a Beatty sequence. Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 523-528. http://geodesic.mathdoc.fr/item/CHEB_2018_19_2_a35/