Article contents
ON THE DIVISOR FUNCTION OVER NONHOMOGENEOUS BEATTY SEQUENCES
Published online by Cambridge University Press: 04 March 2022
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 $
.
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 280 - 287
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
References
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