Generalization of the Lehmer problem over incomplete intervals

Journal of Inequalities and Applications

Research

Di Han¹  Zhaoying Liu¹ 

[1] Research Center for Number Theory and its Applications, Northwest University, Xi’an, P.R. China;

关键词: Lehmer problem;    Generalized Kloosterman sums;    Incomplete interval;    m;   

DOI  :  10.1186/s13660-023-03034-9

 received in 2023-01-16, accepted in 2023-09-15,  发布年份 2023

来源: Springer

【 摘 要 】

Let α≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha \geq 2$\end{document}, m≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$m\geq 2 $\end{document} be integers, p be an odd prime with p∤m(m+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$p\nmid m (m+1 )$\end{document}, 0<λ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$0<\lambda _{1} $\end{document}, λ2≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\lambda _{2}\leq 1$\end{document} be real numbers, q=pα>max{[1λ1],[1λ2]}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$q=p^{\alpha }> \max \{ [ \frac{1}{\lambda _{1}} ], [ \frac{1}{\lambda _{2}} ] \}$\end{document}. For any integer n with (n,q)=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(n,q)=1$\end{document} and a nonnegative integer k, we define Mλ1,λ2(m,n,k;q)=∑′a=1q∑′b=1[λ1q]∑′c=1[λ2q]ab≡1(modq)c≡am(modq)n∤b+c(b−c)2k.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ M_{\lambda _{1},\lambda _{2}} ( m,n,k;q )=\mathop{\mathop{ \mathop{\mathop{{\sum }'}_{a=1}^{q}\mathop{{\sum }'}_{b=1}^{ [ \lambda _{1}q ]}\mathop{{\sum }'}_{c=1}^{ [\lambda _{2}q ]}}_{ab\equiv 1(\bmod q)}}_{c\equiv a^{m}(\bmod q)}}_{n\nmid b+c} ( b-c )^{2k}. $$\end{document} In this paper, we study the arithmetic properties of these generalized Kloosterman sums and give an upper bound estimation for it. By using the upper bound estimation, we discuss the properties of Mλ1,λ2(m,n,k;q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$M_{\lambda _{1},\lambda _{2}} ( m,n,k;q )$\end{document} and obtain an asymptotic formula.

【 授权许可】

CC BY   
© Springer Nature Switzerland AG 2023

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