Generalization of the Lehmer problem over incomplete intervals

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.


Introduction
Let q > 2 be an odd integer.For any integer 1 ≤ a, b ≤ q -1 with (a, q) = 1, there exists a unique integer 1 ≤ c ≤ q -1 such that bc ≡ a(mod q).Let N(a, q) denote the number of solutions of the congruence equation bc ≡ a (mod q) with 2 b + c and (b, q) = (c, q) = 1.The classic D. H. Lehmer problem (see F12 in [2]) is to find some nontrivial properties about N (1, p), where p is an odd prime.Zhang [10] has given an asymptotic estimate In [11], Zhang studied the D. H. Lehmer problem in the general case of an odd number q > 2 and obtained N(1, q) = 1 2 ϕ(q) + O q 1 2 d 2 (q) ln 2 q , where ϕ(q) is the Euler function and d(q) is the divisor function.For further properties of N(a, q), Zhang [15] studied the mean square value of the error term N(a, q) -ϕ(q) 2 , which proved that the estimate in [11] is best possible.Lu and Yi [6] generalized the condition "2 a + b" of the classic Lehmer problem to the general case "n a + b".Let λ 1 , λ 2 be any real number with 0 < λ 1 , λ 2 ≤ 1, n ≥ 2 be a fixed integer, c and q ≥ 3 be integers with (n, q) = (c, q) = 1.They derived n a+b 1 = 1 -1 n λ 1 λ 2 ϕ(q) + O q 1 2 d 6 (q) ln 2 q .
Khan and Shparlinski [4,5] studied the maximal difference between an integer and its inverse M(q) = max |a -ā| : 1 ≤ a ≤ q, (a, q) = 1 and proved for any > 0. Then Xu and Yi [8] generalized the problem in [14], they focused on the distribution behavior of |a -ā| over incomplete intervals.For any real numbers λ, δ with 0 < λ, δ ≤ 1, they studied and gave an asymptotic formula for it.
In addition, the research on the mean value distribution of the difference between an integer and its inverse has also aroused the interest of many scholars.Zhang [13] was the first person to explicitly address this issue, he proved Let (a m ) q represent the minimum positive residue of the integer a m modulo q, that is, 1 ≤ b ≤ q is an integer with b ≡ a m (mod q).Xu [7] generalized the problem in [13] and studied the distribution of the differences |a -(a m ) q | over incomplete intervals [1, [λq]] with 0 < λ ≤ 1.He defined and obtained a sharp asymptotic formula Let ā denote the inverse of a modulo q, and a is called a D. H. Lehmer number if 2 a + ā.
Zhang [12] studied the even power mean of the distance between a and ā with a Lehmer number and proved In 2014, Xu and Zhang [9] considered the high-dimensional case.Let 0 < λ 1 , . . ., λ k+1 ≤ 1 be real numbers, q ≥ max{[ 1 λ i ] : 1 ≤ i ≤ k + 1} be a positive integer with (a, q) = 1.For any nonnegative integer m, they considered the distribution of the 2mth powers of b and obtained an asymptotic formula for Han, Xu, Yi, and Zhang [3] recently generalized the problem in [9].They studied the highdimensional D. H. Lehmer problem and gave an asymptotic formula for it.
In this paper, we generalize the problem in [7,12] and study the difference between ā and (a m ) q with n ā + (a m ) q .Let m ≥ 2, α ≥ 2 be integers, p be an odd prime with p m(m + 1), , n be an integer with (n, q) = 1, k be a nonnegative integer.For any integer a ∈ [1, q) with (a, q) = 1, we define The main purpose of this paper is to study the asymptotic properties of M λ 1 ,λ 2 (m, n, k; q) by using the estimation for the generalized Kloosterman sums and properties of trigonometric sums.
We will prove the following result.
Theorem 1 Let m ≥ 2, α ≥ 2 be integers, p be an odd prime with p m(m + 1), 0 < λ 1 , , n be an integer with (n, q) = 1, k be a nonnegative integer.For any integer a ∈ [1, q) with (a, q) = 1, we have where ϕ(q) is the Euler function and d(q) is the divisor function. Let it is clear that M 1,1 (m, n, 0; q) = F q (m, n), so we can get the following corollary.
Corollary 1 Let m ≥ 2, α ≥ 2 be integers, p be an odd prime with p m(m + 1), q = p α > 2, n be an integer with (n, q) = 1.We have Taking n = 2, we get the following result.

Lemma 3
Let q and i be integers with q > 2, i ≥ 0. Let r and l be integers with 1 ≤ r ≤ q, 1 ≤ l ≤ n.Let 0 < λ ≤ 1 be a real number.For any given integer n ≥ 2, we have Proof See Lemma 2.1 in [3].

Lemma 4
Let p be an odd prime.Then Proof See Lemma 7 in [1].

Lemma 6 Let r, s be integers and p be a prime. For any positive integer m
Proof We have (r, s, p α ) = p h with 0 ≤ h ≤ α.The case h = α is trivial, so we let h < α, a = p β v + u, where β = αh, we can write and we know where p β v + u denotes the integer satisfying 1 ), ū denotes the integer satisfying 1 ≤ ū ≤ p β , u ū ≡ 1(mod p β ).Therefore we have Lemma 7 Let r, s be integers and p be an odd prime.For any positive integers m ≥ 2 and α ≥ 2, p m(m + 1), q = p α , we have S m (r, s, q) = q a=1 e r ā + sa m q ≤ (m + 1)(r, s, q) Proof It follows from Lemmas 5 and 6.
Proof Here we only prove inequalities ( 6) and ( 7), the others can be obtained by the same method.To prove (6), using Lemma 7 and the inequality we can write To prove (7), we can write This proves Lemma 8.
From Lemma 3, we know while the other terms are Using Lemma 8, we know the above formula is (m + 1)q i+j+ 1 2 d(q) ln 2 q.Therefore q a=1 This proves Lemma 9.