$p$-adic formal series and primitive polynomials over finite fields

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over $p$-adic number fields and the estimates of character sums over Galois rings. Given $n$ we prove, for large enough $q$, the Hansen-Mullen conjecture that there exists a primitive polynomial $f(x)=x^{n}-a_{1}x^{n-1}+\cdots +(-1)^{n}a_{n}$ over $F_{q}$ of degree $n$ with the $m$-th ($0<m<n)$ coefficient $a_{m}$ fixed in advance except when $m=\frac{n+1}{2}$ if $n$ is odd and when $m=\frac{n}{2}, \frac{n}{2}+1$ if $n$ is even.