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June, 2022 Consecutive Quadratic Residues and Primitive Roots in the Sequences Formed by Twice-differentiable Functions
Mengyao Jing, Huaning Liu
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Taiwanese J. Math. 26(3): 445-461 (June, 2022). DOI: 10.11650/tjm/211206

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

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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

Information

Received: 14 October 2021; Revised: 3 December 2021; Accepted: 15 December 2021; Published: June, 2022
First available in Project Euclid: 26 December 2021

Digital Object Identifier: 10.11650/tjm/211206

Subjects:
Primary: 11A07 , 11B83 , 11L40

Keywords: character sum , primitive root , Quadratic residue , twice-differentiable function

Rights: Copyright © 2022 The Mathematical Society of the Republic of China

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Vol.26 • No. 3 • June, 2022
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