The classical Wecken theorem claims that any self-map $f:M\to M$ of a compact manifold of dimension $\ge 3$ is homotopic to a map having exactly $N(f)$ fixed points where $N(f)$ denotes the Nielsen number. In 1983 Boju Jiang introduced an algebraically computable number $NF_n(f)$ which is an estimate of the cardinality of {\it n-periodic point set} $\{x\in M; g^n(x)=x\}$ for each $g$ homotopic to $f$.

We prove that every self-map $f:M\to M$ of a compact PL-manifold of dimension $\ge 3$ is homotopic to a map realizing this number i.e. there exists a $g$ homotopic to the given map $f$ and having exactly $NF_n(f)$ n-periodic points. In particular (for $NF_n(f)=0$) the map $f$ is homotopic to map with no n-periodic points iff all Nielsen numbers $N(f^k)$, for all $k$ dividing $n$, disappear.