Abstract
In this paper we bound character sums of the shape \[ \sum_{n \leq N} \chi_1(\lfloor f(n) \rfloor) \chi_2(\lfloor f(n+l) \rfloor), \] where $\chi_1$ and $\chi_2$ are non-principal multiplicative characters modulo a prime $p$, $f(x)$ is a real-valued, twice-differentiable function satisfying a certain condition on $f''(x)$, and $l$ is a positive integer. As an immediate application, we obtain some distribution properties of consecutive quadratic residues and consecutive primitive roots in Piatetski–Shapiro sequences $\lfloor n^c \rfloor$ with $c \in (1,4/3)$.
Funding Statement
This work is supported by National Natural Science Foundation of China under Grant No. 12071368, and the Science and Technology Program of Shaanxi Province of China under Grant Nos. 2019JM-573 and 2020JM-026.
Acknowledgments
The authors express their gratitude to the referee for his/her helpful and detailed comments.
Citation
Mengyao Jing. Huaning Liu. "Consecutive Quadratic Residues and Primitive Roots in the Sequences Formed by Twice-differentiable Functions." Taiwanese J. Math. 26 (3) 445 - 461, June, 2022. https://doi.org/10.11650/tjm/211206
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