On the difference between a D. H. Lehmer number and its inverse over short interval

Abstract

Let $q>2$ be an odd integer. For each integer $x$ with $0<x<q$ and $(q,x)= 1$, we know that there exists one and only one $\bar{x}$ with $0<\bar{x}<q$ such that $x\bar{x}\equiv1(\bmod q)$. A Lehmer number is defined to be any integer $a$ with $2\dagger(a+\bar{a})$. For any nonnegative integer $k$, Let $$ M(x,q,k)=\displaystyle\mathop {\displaystyle\mathop{\sum{'}}_{a=1}^{q} \displaystyle\mathop{\sum{'}}_{b\leq xq}}_{\mbox{$\tiny\begin{array}{c} 2|a+b+1\\ ab\equiv1(\bmod q)\end{array}$}}(a-b)^{2k}.$$ The main purpose of this paper is to study the properties of $M(x,q,k)$, and give a sharp asymptotic formula, by using estimates of Kloosterman's sums and properties of trigonometric sums.