Generalization of the Lehmer problem over incomplete intervals

Abstract

Abstract Let α ≥ 2 $\alpha \geq 2$ , m ≥ 2 $m\geq 2 $ be integers, p be an odd prime with p ∤ m ( m + 1 ) $p\nmid m (m+1 )$ , 0 max { [ 1 λ 1 ] , [ 1 λ 2 ] } $q=p^{\alpha }> \max \{ [ \frac{1}{\lambda _{1}} ], [ \frac{1}{\lambda _{2}} ] \}$ . For any integer n with ( n , q ) = 1 $(n,q)=1$ and a nonnegative integer k, we define M λ 1 , λ 2 ( m , n , k ; q ) = ∑ ′ a = 1 q ∑ ′ b = 1 [ λ 1 q ] ∑ ′ c = 1 [ λ 2 q ] a b ≡ 1 ( mod q ) c ≡ a m ( mod q ) n ∤ b + c ( b − c ) 2 k . $$ M_{\lambda _{1},\lambda _{2}} ( m,n,k;q )=\mathop{\mathop{ \mathop{\mathop{{\sum }'}_{a=1}^{q}\mathop{{\sum }'}_{b=1}^{ [ \lambda _{1}q ]}\mathop{{\sum }'}_{c=1}^{ [\lambda _{2}q ]}}_{ab\equiv 1(\bmod q)}}_{c\equiv a^{m}(\bmod q)}}_{n\nmid b+c} ( b-c )^{2k}. $$ 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 λ 1 , λ 2 ( m , n , k ; q ) $M_{\lambda _{1},\lambda _{2}} ( m,n,k;q )$ and obtain an asymptotic formula.

Keywords

• Lehmer problem
• Generalized Kloosterman sums
• Incomplete interval
• mth power q