On the divisor function over Piatetski-Shapiro sequences

Hui Wang; Yu Zhang

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 2, page 613-620
  • ISSN: 0011-4642

Abstract

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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 < 6 5 , ∑ n ≤ x d ( [ n c ] ) = c x log x + ( 2 γ - c ) x + O x log x , where γ is the Euler constant and [ n c ] is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

How to cite

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Wang, Hui, and Zhang, Yu. "On the divisor function over Piatetski-Shapiro sequences." Czechoslovak Mathematical Journal 73.2 (2023): 613-620. <http://eudml.org/doc/299597>.

@article{Wang2023,
abstract = {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 $1<c<\frac\{6\}\{5\}$, \[ \sum \_\{n\le x\} d([n^c])= cx\log x +(2\gamma -c)x+O\Bigl (\frac\{x\}\{\log x\}\Bigr ), \] 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.},
author = {Wang, Hui, Zhang, Yu},
journal = {Czechoslovak Mathematical Journal},
keywords = {divisor function; Piatetski-Shapiro sequence; exponential sum},
language = {eng},
number = {2},
pages = {613-620},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the divisor function over Piatetski-Shapiro sequences},
url = {http://eudml.org/doc/299597},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Wang, Hui
AU - Zhang, Yu
TI - On the divisor function over Piatetski-Shapiro sequences
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 613
EP - 620
AB - 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 $1<c<\frac{6}{5}$, \[ \sum _{n\le x} d([n^c])= cx\log x +(2\gamma -c)x+O\Bigl (\frac{x}{\log x}\Bigr ), \] 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.
LA - eng
KW - divisor function; Piatetski-Shapiro sequence; exponential sum
UR - http://eudml.org/doc/299597
ER -

References

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  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. (1999) Zbl0983.11055MR1735145
  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. (1999) Zbl0957.11040MR1706007
  3. Graham, S. W., Kolesnik, G., 10.1017/CBO9780511661976, London Mathematical Society Lecture Note Series 126. Cambridge University Press, Cambridge (1991). (1991) Zbl0713.11001MR1145488DOI10.1017/CBO9780511661976
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  5. Jutila, M., Lectures on a Method in the Theory of Exponential Sums, Tata Institute Lectures on Mathematics and Physics 80. Springer, Berlin (1987). (1987) Zbl0671.10031MR0910497
  6. Kolesnik, G. A., 10.1007/BF01101405, Math. Notes 6 (1969), 784-791 translation from Mat. Zametki 6 1969 545-554. (1969) Zbl0233.10031MR0257004DOI10.1007/BF01101405
  7. Kolesnik, G. A., 10.2140/PJM.1985.118.437, Pac. J. Math. 118 (1985), 437-447. (1985) Zbl0571.10037MR0789183DOI10.2140/PJM.1985.118.437
  8. Liu, H. Q., Rivat, J., 10.1112/BLMS/24.2.143, Bull. Lond. Math. Soc. 24 (1992), 143-147. (1992) Zbl0772.11032MR1148674DOI10.1112/BLMS/24.2.143
  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. (2003) Zbl1481.11090MR2058014
  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. (1953) Zbl0053.02702MR0059302
  11. Rivat, J., Sargos, P., 10.4153/CJM-2001-017-0, Can. J. Math. 53 (2001), 414-433 French. (2001) Zbl0970.11035MR1820915DOI10.4153/CJM-2001-017-0
  12. Rivat, J., Wu, J., 10.1017/S0017089501020080, Glasg. Math. J. 43 (2001), 237-254. (2001) Zbl0987.11052MR1838628DOI10.1017/S0017089501020080
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