ON THE DIVISOR FUNCTION OVER NONHOMOGENEOUS BEATTY SEQUENCES

We consider sums involving the divisor function over nonhomogeneous ( $\beta \neq 0$ ) Beatty sequences $ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta ]\}_{n=1}^{\infty } $ and show that $$ \begin{align*} \sum_{n\leq N,\ n\in\mathcal{B}_{\alpha,\beta}}d(n) =\alpha^{-1}\sum_{m\leq N}d(m) +O(N^{1-...

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Bibliographic Details
Published inBulletin of the Australian Mathematical Society Vol. 106; no. 2; pp. 280 - 287
Main Author ZHANG, WEI
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.10.2022
Subjects
Estimates
Mathematical functions
Sequences
divisor problems
exponential sums
11L07
11L20
11B83
Beatty sequences
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Summary:We consider sums involving the divisor function over nonhomogeneous ( $\beta \neq 0$ ) Beatty sequences $ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta ]\}_{n=1}^{\infty } $ and show that $$ \begin{align*} \sum_{n\leq N,\ n\in\mathcal{B}_{\alpha,\beta}}d(n) =\alpha^{-1}\sum_{m\leq N}d(m) +O(N^{1-1/(\tau+1)+\varepsilon}), \end{align*} $$ where N is a sufficiently large integer, $\alpha $ is of finite type $\tau $ and $\beta \neq 0$ . Previously, such estimates were only obtained for homogeneous Beatty sequences or for almost all $\alpha $ .
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ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972722000181