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

A. V. Begunts, D. V. Goryashin

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

UDC: 511.35, 517.15

Received: 19.06.2018
Accepted: 17.08.2018

DOI: 10.22405/2226-8383-2018-19-2-523-528

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