| Title:
|
On the distribution of $(k,r)$-integers in Piatetski-Shapiro sequences (English) |
| Author:
|
Srichan, Teerapat |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
1063-1070 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A natural number $n$ is said to be a $(k,r)$-integer if $n=a^kb$, where $k>r>1$ and $b$ is not divisible by the $r$th power of any prime. We study the distribution of such $(k,r)$-integers in the Piatetski-Shapiro sequence $\{\lfloor n^c \rfloor \}$ with $c>1$. As a corollary, we also obtain similar results for semi-$r$-free integers. (English) |
| Keyword:
|
$(k,r)$-integer |
| Keyword:
|
Piatetski-Shapiro sequence |
| MSC:
|
11L07 |
| MSC:
|
11N37 |
| idZBL:
|
Zbl 07442474 |
| idMR:
|
MR4339111 |
| DOI:
|
10.21136/CMJ.2021.0194-20 |
| . |
| Date available:
|
2021-11-08T16:00:55Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149238 |
| . |
| Reference:
|
[1] Cao, X., Zhai, W.: The distribution of square-free numbers of the form $[n^c]$.J. Théor. Nombres Bordx. 10 (1998), 287-299. Zbl 0926.11066, MR 1828246, 10.5802/jtnb.229 |
| Reference:
|
[2] Cao, X., Zhai, W.: On the distribution of square-free numbers of the form $[n^c]$. II.Acta Math. Sin., Chin. Ser. 51 (2008), 1187-1194 Chinese. Zbl 1174.11395, MR 2490038 |
| Reference:
|
[3] Deshouillers, J.-M.: A remark on cube-free numbers in Segal-Piatetski-Shapiro sequences.Hardy-Ramanujan J. 41 (2018), 127-132. Zbl 1448.11055, MR 3935505 |
| Reference:
|
[4] Ivić, A.: The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function.A Wiley-Interscience Publication. John Wiley & Sons, New York (1985). Zbl 0556.10026, MR 792089 |
| Reference:
|
[5] Piatetski-Shapiro, I. I.: On the distribution of prime numbers in sequences of the form $[f(n)]$.Mat. Sb., N.Ser. 33 (1953), 559-566 Russian. Zbl 0053.02702, MR 0059302 |
| Reference:
|
[6] Rieger, G. J.: Remark on a paper of Stux concerning squarefree numbers in non-linear sequences.Pac. J. Math. 78 (1978), 241-242. Zbl 0395.10049, MR 513296, 10.2140/pjm.1978.78.241 |
| Reference:
|
[7] Stux, I. E.: Distribution of squarefree integers in non-linear sequences.Pac. J. Math. 59 (1975), 577-584. Zbl 0297.10033, MR 387218, 10.2140/pjm.1975.59.577 |
| Reference:
|
[8] Subbarao, M. V., Suryanarayana, D.: On the order of the error function of the $(k,r)$integers.J. Number Theory 6 (1974), 112-123. Zbl 0277.10036, MR 0335454, 10.1016/0022-314X(74)90049-3 |
| Reference:
|
[9] Zhang, M., Li, J.: Distribution of cube-free numbers with form $[n^c]$.Front. Math. China 12 (2017), 1515-1525. Zbl 1418.11137, MR 3722230, 10.1007/s11464-017-0652-1 |
| . |
