In this paper, we study the number of the Lehmer primitive roots solutions of a multivariate linear equation and the number of $1\le x\le p-1$ such that for $f(x)\in {\mathbb{F}}_p[x]$, $k$ polynomials $f(x+c_1),f(x+c_2),\dots ,f(x+c_k)$ are Lehmer primitive roots modulo prime $p$, and obtain asymptotic formulae for these utilizing the properties of Gauss sums and the generalized Kloosterman sums.

The main purpose of this paper is to use elementary and analytic methods to study the calculating problem of one kind of Gauss sums and obtain an exact computational formula for it.