Geodesic
On the divisor function over Piatetski-Shapiro sequences
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 613-620
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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
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 $1$, $$ \sum _{n\leq x} d([n^c])= cx\log x +(2\gamma -c)x+O\Bigl (\frac {x}{\log x}\Bigr ), $$ 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.
DOI : 10.21136/CMJ.2023.0205-22
Classification : 11B83, 11L07, 11N25, 11N37
Keywords: divisor function; Piatetski-Shapiro sequence; exponential sum 
@article{10_21136_CMJ_2023_0205_22,
     author = {Wang, Hui and Zhang, Yu},
     title = {On the divisor function over {Piatetski-Shapiro} sequences},
     journal = {Czechoslovak Mathematical Journal},
     pages = {613--620},
     year = {2023},
     volume = {73},
     number = {2},
     doi = {10.21136/CMJ.2023.0205-22},
     mrnumber = {4586914},
     zbl = {07729527},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0205-22/}
}
TY  - JOUR
AU  - Wang, Hui
AU  - Zhang, Yu
TI  - On the divisor function over Piatetski-Shapiro sequences
JO  - Czechoslovak Mathematical Journal
PY  - 2023
SP  - 613
EP  - 620
VL  - 73
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0205-22/
DO  - 10.21136/CMJ.2023.0205-22
LA  - en
ID  - 10_21136_CMJ_2023_0205_22
ER  - 
%0 Journal Article
%A Wang, Hui
%A Zhang, Yu
%T On the divisor function over Piatetski-Shapiro sequences
%J Czechoslovak Mathematical Journal
%D 2023
%P 613-620
%V 73
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0205-22/
%R 10.21136/CMJ.2023.0205-22
%G en
%F 10_21136_CMJ_2023_0205_22
Wang, Hui; Zhang, Yu. On the divisor function over Piatetski-Shapiro sequences. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 613-620. doi: 10.21136/CMJ.2023.0205-22

[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. | MR | JFM

[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. | MR | JFM

[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). | DOI | MR | JFM

[4] Heath-Brown, D. R.: The Pjateckii-Šapiro prime number theorem. J. Number Theory 16 (1983), 242-266. | DOI | MR | JFM

[5] Jutila, M.: Lectures on a Method in the Theory of Exponential Sums. Tata Institute Lectures on Mathematics and Physics 80. Springer, Berlin (1987). | MR | JFM

[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. | DOI | MR | JFM

[7] Kolesnik, G. A.: Primes of the form $[n^c]$. Pac. J. Math. 118 (1985), 437-447. | DOI | MR | JFM

[8] Liu, H. Q., Rivat, J.: On the Pjateckii-Šapiro prime number theorem. Bull. Lond. Math. Soc. 24 (1992), 143-147. | DOI | MR | JFM

[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. | MR | JFM

[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. | MR | JFM

[11] Rivat, J., Sargos, P.: Nombres premiers de la forme $[n^c]$. Can. J. Math. 53 (2001), 414-433 French. | DOI | MR | JFM

[12] Rivat, J., Wu, J.: Prime numbers of the form $[n^c]$. Glasg. Math. J. 43 (2001), 237-254. | DOI | MR | JFM

[13] Vaaler, J. D.: Some extremal functions in Fourier analysis. Bull. Am. Math. Soc., New Ser. 12 (1985), 183-216. | DOI | MR | JFM

Cité par Sources :