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 $xbar{x}equiv1(bmod q)$. A Lehmer number is defined to be any integer $a$ with $2dagger(a+bar{a})$. For any nonnegative integer $k$, Let $$ M(x,q,k)=displaystylemathop {displaystylemathop{sum{}}_{a=1}^{q} displaystylemathop{sum{}}_{bleq xq}}_{mbox{$tinybegin{array}{c} 2|a+b+1 abequiv1(bmod q)end{array}$}}(a-b)^{2k}.$$ 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 Kloostermans sums and properties of trigonometric sums.
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