Some new sums related to D. H. Lehmer problem

summary:About Lehmer's number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let $p$ be a prime, and let $N(k; p)$ denote the number of all $1 \leq a_i \leq p - 1 $ such that $a_1a_2 \cdots a_k \equiv 1 \mod p$ and $2 \mid a_i + \bar {a}_i + 1,$ $i = 1, 2, \cdots , k$. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function $N(k; p),$ and give an interesting asymptotic formula for it