Generalization of the Lehmer problem over incomplete intervals

Abstract

Let alpha geq 2, mgeq 2 be integers, p be an odd prime with pnmid m (m+1 ), 0<lambda _{1} , lambda _{2}leq 1 be real numbers, q=p^{alpha }> max { [ frac{1}{lambda _{1}} ], [ frac{1}{lambda _{2}} ] }. For any integer n with (n,q)=1 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}$$ M_{\\lambda _{1},\\lambda _{2}} ( m,n,k;q )=\\mathop{\\mathop{ \\mathop{\\mathop{{\\sum }'}_{a=1}^{q}\\mathop{{\\sum }'}_{b=1}^{ [ \\lambda _{1}q ]}\\mathop{{\\sum }'}_{c=1}^{ [\\lambda _{2}q ]}}_{ab\\equiv 1(\\bmod q)}}_{c\\equiv a^{m}(\\bmod q)}}_{n\ mid b+c} ( b-c )^{2k}. $$\\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 M_{lambda _{1},lambda _{2}} ( m,n,k;q ) and obtain an asymptotic formula.