Article
| Title: | On the divisor function over Piatetski-Shapiro sequences (English) |
| Author: | Wang, Hui |
| Author: | Zhang, Yu |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 2 |
| Year: | 2023 |
| Pages: | 613-620 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $[x]$ be an integer part of $x$ and $d(n)$ be the number of positive divisor of $n$. Inspired by some results of M. Jutila (1987), we prove that for $1<c<\frac 65$, $$ \sum _{n\leq x} d([n^c])= cx\log x +(2\gamma -c)x+O\Bigl (\frac {x}{\log x}\Bigr ), $$ where $\gamma $ is the Euler constant and $[n^c]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem. (English) |
| Keyword: | divisor function |
| Keyword: | Piatetski-Shapiro sequence |
| Keyword: | exponential sum |
| MSC: | 11B83 |
| MSC: | 11L07 |
| MSC: | 11N25 |
| MSC: | 11N37 |
| idZBL: | Zbl 07729527 |
| idMR: | MR4586914 |
| DOI: | 10.21136/CMJ.2023.0205-22 |
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| Date available: | 2023-05-04T17:51:44Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151677 |
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| Reference: | [8] Liu, H. Q., Rivat, J.: On the Pjateckii-Šapiro prime number theorem.Bull. Lond. Math. Soc. 24 (1992), 143-147. Zbl 0772.11032, MR 1148674, 10.1112/BLMS/24.2.143 |
| Reference: | [9] Lü, G. S., Zhai, W. G.: The sum of multidimensional divisor functions on a special sequence.Adv. Math., Beijing 32 (2003), 660-664 Chinese. Zbl 1481.11090, MR 2058014 |
| Reference: | [10] Piatetski-Shapiro, I. I.: On the distribution of the prime numbers in sequences of the form $[f(n)]$.Mat. Sb., N. Ser. 33 (1953), 559-566 Russian. Zbl 0053.02702, MR 0059302 |
| Reference: | [11] Rivat, J., Sargos, P.: Nombres premiers de la forme $[n^c]$.Can. J. Math. 53 (2001), 414-433 French. Zbl 0970.11035, MR 1820915, 10.4153/CJM-2001-017-0 |
| Reference: | [12] Rivat, J., Wu, J.: Prime numbers of the form $[n^c]$.Glasg. Math. J. 43 (2001), 237-254. Zbl 0987.11052, MR 1838628, 10.1017/S0017089501020080 |
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