Let $ q $ be a sufficiently large odd integer, and let $ c \in\left(1, \frac{4}{3}\right) $. We denote $ R(c; q) $ as the count of square-free numbers in the intersection of the Lehmer set and the Piatetski-Shapiro sequence. By employing additive character properties to transform congruence equations and applying Kloosterman sums and methods of exponential sums, we derive a sharp asymptotic formula as $ q $ approaches infinity, which is significant for understanding the distribution properties of the Lehmer problem.
Citation: Zhao Xiaoqing, Yi Yuan. Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence[J]. AIMS Mathematics, 2024, 9(12): 33591-33609. doi: 10.3934/math.20241603
Let $ q $ be a sufficiently large odd integer, and let $ c \in\left(1, \frac{4}{3}\right) $. We denote $ R(c; q) $ as the count of square-free numbers in the intersection of the Lehmer set and the Piatetski-Shapiro sequence. By employing additive character properties to transform congruence equations and applying Kloosterman sums and methods of exponential sums, we derive a sharp asymptotic formula as $ q $ approaches infinity, which is significant for understanding the distribution properties of the Lehmer problem.
| [1] | T. M. Apostol, Introduction to analytic number theory, Springer-Verlag, 1976. https://doi.org/10.1007/978-1-4757-5579-4 |
| [2] | R. K. Guy, Unsolved problems in number theory, 3 Eds., Springer-Verlag, 2004. https://doi.org/10.1007/978-0-387-26677-0 |
| [3] | W. Zhang, A problem of D. H. Lehmer and its generalization (Ⅱ), Compos. Math., 91 (1994), 47–56. |
| [4] | W. Zhang, A problem of D. H. Lehmer and its generalization, Compos. Math., 86 (1993), 307–316. |
| [5] |
Y. Lu, Y. Yi, On the generalization of the D. H. Lehmer problem, Acta Math. Sin. Engl. Ser., 25 (2009), 1269–1274. https://doi.org/10.1007/s10114-009-7652-3 doi: 10.1007/s10114-009-7652-3
|
| [6] |
P. Xi, Y. Yi, Generalized D. H. Lehmer problem over short intervals, Glasg. Math. J., 53 (2011), 293–299. https://doi.org/10.1017/S0017089510000704 doi: 10.1017/S0017089510000704
|
| [7] |
Z. Liu, D. Han, Generalization of the Lehmer problem over incomplete intervals, J. Inequal. Appl., 2023 (2023), 128. https://doi.org/10.1186/s13660-023-03034-9 doi: 10.1186/s13660-023-03034-9
|
| [8] | V. Z. Guo, Y. Yi, The Lehmer problem and Beatty sequences, Bull. Math. Soc. Sci. Math. Roum., 67 (2024), 471–482. |
| [9] | I. I. Pyateckiĭ-Šapiro, On the distribution of prime numbers in sequences of the form [$f(n)$], Mat. Sb., 33 (1953), 559–566. |
| [10] |
I. E. Stux, Distribution of squarefree integers in non-linear sequences, Pacific J. Math., 59 (1975), 577–584. https://doi.org/10.2140/PJM.1975.59.577 doi: 10.2140/PJM.1975.59.577
|
| [11] |
G. J. Rieger, Remark on a paper of Stux concerning squarefree numbers in non-linear sequences, Pacific J. Math., 78 (1978), 241–242. http://doi.org/10.2140/pjm.1978.78.241 doi: 10.2140/pjm.1978.78.241
|
| [12] | S. W. Graham, G. Kolesnik, Van der Corput's method of exponential sums, Cambridge University Press, 1991. https://doi.org/10.1017/CBO9780511661976 |
| [13] |
T. Estermann, On Kloosterman's sum, Mathematika, 8 (1961), 83–86. https://doi.org/10.1112/S0025579300002187 doi: 10.1112/S0025579300002187
|
| [14] | N. M. Korobov, Exponential sums and their applications, Springer-Verlag, 1992. https://doi.org/https://doi.org/10.1007/978-94-015-8032-8 |
| [15] | A. A. Karatsuba, M. B. Nathanson, Basic analytic number theory, 2 Eds., Springer-Verlag, 1993. https://doi.org/10.1007/978-3-642-58018-5 |
Zhao Xiaoqing, Yi Yuan. Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence[J]. AIMS Mathematics, 2024, 9(12): 33591-33609. doi: 10.3934/math.20241603
DownLoad: