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 $$ \begin{align*} \sum_{n\leq N,\ n\in\mathcal{B}_{\alpha,\beta}}d(n) =\alpha^{-1}\sum_{m\leq N}d(m) +O(N^{1-...
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| Published in | Bulletin of the Australian Mathematical Society Vol. 106; no. 2; pp. 280 - 287 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge, UK
Cambridge University Press
01.10.2022
|
| Subjects | |
| Online Access | Get full text |
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| Abstract | We consider sums involving the divisor function over nonhomogeneous (
$\beta \neq 0$
) Beatty sequences
$ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta ]\}_{n=1}^{\infty } $
and show that
$$ \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*} $$
where N is a sufficiently large integer,
$\alpha $
is of finite type
$\tau $
and
$\beta \neq 0$
. Previously, such estimates were only obtained for homogeneous Beatty sequences or for almost all
$\alpha $
. |
|---|---|
| AbstractList | We consider sums involving the divisor function over nonhomogeneous (
$\beta \neq 0$
) Beatty sequences
$ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta ]\}_{n=1}^{\infty } $
and show that
$$ \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*} $$
where
N
is a sufficiently large integer,
$\alpha $
is of finite type
$\tau $
and
$\beta \neq 0$
. Previously, such estimates were only obtained for homogeneous Beatty sequences or for almost all
$\alpha $
. We consider sums involving the divisor function over nonhomogeneous ( $\beta \neq 0$ ) Beatty sequences $ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta ]\}_{n=1}^{\infty } $ and show that $$ \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*} $$ where N is a sufficiently large integer, $\alpha $ is of finite type $\tau $ and $\beta \neq 0$ . Previously, such estimates were only obtained for homogeneous Beatty sequences or for almost all $\alpha $ . We consider sums involving the divisor function over nonhomogeneous ($\beta \neq 0$) Beatty sequences $ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta ]\}_{n=1}^{\infty } $ and show that $$ \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 $. |
| Author | ZHANG, WEI |
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| References | 1926; 24 2021; 197 1995; 70 1977; 24 2006; 13 1931; 33 1997; 42 1948; 12 2009; 136 2004; 47 1955; 2 S0004972722000181_r1 Zhai (S0004972722000181_r14) 1997; 42 S0004972722000181_r2 S0004972722000181_r11 S0004972722000181_r3 S0004972722000181_r10 S0004972722000181_r4 S0004972722000181_r12 Vinogradov (S0004972722000181_r13) 2004 Kuipers (S0004972722000181_r8) 1974 Iwaniec (S0004972722000181_r6) 2004 Erdős (S0004972722000181_r5) 1948; 12 Lü (S0004972722000181_r9) 2004; 47 S0004972722000181_r7 |
| References_xml | – volume: 13 start-page: 539 year: 2006 end-page: 547 article-title: Short character sums with Beatty sequences publication-title: Math. Res. Lett. – volume: 136 start-page: 81 year: 2009 end-page: 89 article-title: Arithmetic functions on Beatty sequences publication-title: Acta Arith. – volume: 47 start-page: 1213 year: 2004 end-page: 1216 article-title: The divisor problem for the Beatty sequences publication-title: Acta Math. Sinica (Chin. Ser.) – volume: 197 start-page: 93 year: 2021 end-page: 104 article-title: Metric results on summatory arithmetic functions on Beatty sets publication-title: Acta Arith. – volume: 42 start-page: 804 year: 1997 end-page: 806 article-title: Note on a result of Abercrombie publication-title: Chinese Sci. Bull. – volume: 2 start-page: 1 year: 1955 end-page: 20 article-title: Rational approximations to algebraic numbers publication-title: Mathematika – volume: 70 start-page: 195 year: 1995 end-page: 207 article-title: Beatty sequences and multiplicative number theory publication-title: Acta Arith. – volume: 33 start-page: 544 year: 1931 end-page: 563 article-title: Some problems of Diophantine approximation I publication-title: Math. Z. – volume: 24 start-page: 706 year: 1926 end-page: 714 article-title: Zur metrischen Theorie der diophantischen Approximationen publication-title: Math. Z. – volume: 12 start-page: 67 year: 1948 end-page: 74 article-title: Some remarks on Diophantine approximations publication-title: J. Indian Math. Soc. (N.S.) – volume: 24 start-page: 135 year: 1977 end-page: 141 article-title: modulo 1 publication-title: Mathematika – ident: S0004972722000181_r11 doi: 10.4064/aa200128-10-6 – ident: S0004972722000181_r1 doi: 10.4064/aa-70-3-195-207 – ident: S0004972722000181_r10 doi: 10.1112/S0025579300000644 – volume: 47 start-page: 1213 year: 2004 ident: S0004972722000181_r9 article-title: The divisor problem for the Beatty sequences publication-title: Acta Math. Sinica (Chin. Ser.) – volume-title: The Method of Trigonometrical Sums in the Theory of Numbers year: 2004 ident: S0004972722000181_r13 – volume: 12 start-page: 67 year: 1948 ident: S0004972722000181_r5 article-title: Some remarks on Diophantine approximations publication-title: J. Indian Math. Soc. (N.S.) – volume-title: Uniform Distribution of Sequences year: 1974 ident: S0004972722000181_r8 – ident: S0004972722000181_r4 doi: 10.1007/BF01174369 – ident: S0004972722000181_r12 doi: 10.1112/S0025579300009025 – volume-title: Analytic Number Theory year: 2004 ident: S0004972722000181_r6 – volume: 42 start-page: 804 year: 1997 ident: S0004972722000181_r14 article-title: Note on a result of Abercrombie publication-title: Chinese Sci. Bull. – ident: S0004972722000181_r2 doi: 10.4064/aa136-1-6 – ident: S0004972722000181_r3 doi: 10.4310/MRL.2006.v13.n4.a4 – ident: S0004972722000181_r7 doi: 10.1007/BF01216806 |
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| Snippet | We consider sums involving the divisor function over nonhomogeneous (
$\beta \neq 0$
) Beatty sequences
$ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta... We consider sums involving the divisor function over nonhomogeneous ($\beta \neq 0$) Beatty sequences $ \mathcal {B}_{\alpha ,\beta }:=\{[\alpha n+\beta... |
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| SubjectTerms | Estimates Mathematical functions Sequences |
| Title | ON THE DIVISOR FUNCTION OVER NONHOMOGENEOUS BEATTY SEQUENCES |
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