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ON THE DIVISOR FUNCTION OVER NONHOMOGENEOUS BEATTY SEQUENCES

Part of: Sequences and sets Exponential sums and character sums

Published online by Cambridge University Press:  04 March 2022

WEI ZHANG Open the ORCID record for WEI ZHANG [Opens in a new window]
WEI ZHANG*
Affiliation:
School of Mathematics and Statistics, Henan University, Kaifeng 475004, Henan, PR China
*
e-mail: zhangweimath@126.com
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Abstract

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 $ .

Keywords

exponential sumsBeatty sequencesdivisor problems

MSC classification

Primary: 11L20: Sums over primes
Secondary: 11L07: Estimates on exponential sums 11B83: Special sequences and polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 280 - 287
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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