| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2023-05-04T17:51:44Z |
| Last updated:
|
2025-07-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151677 |
| . |
| Reference:
|
[1] Arkhipov, G. I., Chubarikov, V. N.: On the distribution of prime numbers in a sequence of the form $[n^c]$.Mosc. Univ. Math. Bull. 54 (1999), 25-35 translation from Vestn. Mosk. Univ., Ser. I 1999 1999 25-35. Zbl 0983.11055, MR 1735145 |
| Reference:
|
[2] Arkhipov, G. I., Soliba, K. M., Chubarikov, V. N.: On the sum of values of a multidimensional divisor function on a sequence of type $[n^c]$.Mosc. Univ. Math. Bull. 54 (1999), 28-36 translation from Vestn. Mosk. Univ., Ser. I 1999 1999 28-35. Zbl 0957.11040, MR 1706007 |
| Reference:
|
[3] Graham, S. W., Kolesnik, G.: Van der Corput's Method of Exponential Sums.London Mathematical Society Lecture Note Series 126. Cambridge University Press, Cambridge (1991). Zbl 0713.11001, MR 1145488, 10.1017/CBO9780511661976 |
| Reference:
|
[4] Heath-Brown, D. R.: The Pjateckii-Šapiro prime number theorem.J. Number Theory 16 (1983), 242-266. Zbl 0513.10042, MR 0698168, 10.1016/0022-314X(83)90044-6 |
| Reference:
|
[5] Jutila, M.: Lectures on a Method in the Theory of Exponential Sums.Tata Institute Lectures on Mathematics and Physics 80. Springer, Berlin (1987). Zbl 0671.10031, MR 0910497 |
| Reference:
|
[6] Kolesnik, G. A.: An improvement of the remainder term in the divisor problem.Math. Notes 6 (1969), 784-791 translation from Mat. Zametki 6 1969 545-554. Zbl 0233.10031, MR 0257004, 10.1007/BF01101405 |
| Reference:
|
[7] Kolesnik, G. A.: Primes of the form $[n^c]$.Pac. J. Math. 118 (1985), 437-447. Zbl 0571.10037, MR 0789183, 10.2140/PJM.1985.118.437 |
| 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 |
| Reference:
|
[13] Vaaler, J. D.: Some extremal functions in Fourier analysis.Bull. Am. Math. Soc., New Ser. 12 (1985), 183-216. Zbl 0575.42003, MR 0776471, 10.1090/S0273-0979-1985-15349-2 |
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