Suphawan Janphaisaeng, Teerapat Srichan and Pinthira Tangsupphathawat

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 3, Pages 500–506

DOI: 10.7546/nntdm.2022.28.3.500-506

**Download PDF (170 Kb)**

## Details

### Authors and affiliations

Suphawan Janphaisaeng

*Department of Mathematics, Faculty of Science, Naresuan University
Phitsanulok 65000, Thailand*

Teerapat Srichan

*Department of Mathematics, Faculty of Science, Kasetsart University
Bangkok 10900, Thailand*

Pinthira Tangsupphathawat

*Department of Mathematics, Faculty of Science and Technology, Phranakorn Rajabhat University
Bangkok 10220, Thailand*

### Abstract

In this paper, we study asymptotic behaviour of the sum where under three different types of assumptions on and .

### Keywords

- Arithmetic function, Piatetski-Shapiro sequences

### 2020 Mathematics Subject Classification

- 11N37, 11N69

### References

- Akbal, Y. (2017). Friable values of Piatetski-Shapiro sequences.
*Proceedings of the American Mathematical Society*, 145, 4255–4268. - Baker, R. C., & Banks, W. D. (2015). Character sums with Piatetski-Shapiro sequences.
*The Quarterly Journal of Mathematics*, 66, 393–416. - Baker, R. C., Banks, W. D., Brüdern, J., Shparlinski, I. E., & Weingartner, A. (2013). Piatetski-Shapiro sequences.
*Acta Arithmetica*, 157, 37–68. - Banks, W. D., Guo, V. Z., & Shparlinski, I. E. (2016). Almost primes of the form .
*Indagationes Mathematicae*, 27, 423–436. - Bordellés, O., Dai, L., Heyman, R., Pan, H., & Shparlinski, I. E. (2019). On a sum involving the Euler function. Journal of Number Theory, 202, 278–297.
- Cao, X., & Zhai, W. (1998). The distribution of square-free numbers of the form .
*Journal de Theorie des Nombres de Bordeaux*, 10(2), 287–299. - Deshouillers, J. M. (2019). A remark on cube-free numbers in Segal–Piatestki-Shapiro sequences.
*Hardy–Ramanujan Journal*, 41, 127–132. - Ivić, A. (1985)
*The Riemann Zeta-function: The Theory of the Riemann Zeta-function with Applications*. John Wiley & Sons, Inc., New York. - Liu, K., Shparlinski, I. E., & Zhang, T. (2017). Squares in Piatetski-Shapiro sequences.
*Acta Arithmetica*, 181, 239–252. - Ma, J., Wu, J., & Zhao, F. (2021). On a generalisation of Bordellés–Dai–Heyman–Pan–Shparlinski’s conjecture.
*Journal of Number Theory*, 236, 334–348. - Piatetski-Shapiro, I. I. (1953). On the distribution of prime numbers in sequences of the form .
*Math. Sbornik*, 33, 559–566. - Rieger, G. (1978). Remark on a paper of Stux concerning squarefree numbers in non-linear sequences.
*Pacific Journal of Mathematics*, 78(1), 241–242. - Srichan, T. (2021). On the Distribution of (
*k*,*r*)-Integers in Piatetski-Shapiro Sequences.*Czechoslovak Mathematical Journal*, 71(4), 1063–1070. - Srichan, T. Multiplicative functions of special type on Piatetski-Shapiro sequences.
*Mathematica Slovaca*(to appear). - Srichan, T., & Tangsupphathawat, P. (2020). Square-full numbers in Piatetski-Shapiro sequences.
*Annales mathématiques du Québec*, 44(2), 385–391. - Stux, I. (1975). Distribution of squarefree integers in non-linear sequences.
*Pacific Journal of Mathematics*, 59(2), 577–584. - Wu, J. (2020). Note on a paper by Bordellés, Dai, Heyman, Pan and Shparlinski.
*Periodica Mathematica Hungarica*, 80, 95–102. - Zhai, W. (2020). On a sum involving the Euler function.
*Journal of Number Theory*, 211, 199–219. - Zhang, M., & Jinjiang, L. (2017). Distribution of cube-free numbers with form .
*Frontiers of Mathematics in China*, 12(6), 1515–1525. - Zhao, F., & Wu, J. (2022). Note on a paper by Bordellés, Dai, Heyman, Pan and Shparlinski, II.
*Acta Arithmetica*, 202, 185–194.

### Manuscript history

- Received: 2 March 2022
- Revised: 25 July 2022
- Accepted: 2 August 2022
- Online First: 4 August 2022

## Related papers

## Cite this paper

Janphaisaeng, S., Srichan, T., & Tangsupphathawat, P. (2022). Average value of some certain types of arithmetic functions with Piatetski-Shapiro sequences. *Notes on Number Theory and Discrete Mathematics*, 28(3), 500-506, DOI: 10.7546/nntdm.2022.28.3.500-506.