4. Let $x_{n}=\sum_{k=1}^{2013}\left(\cos \frac{k!\pi}{2013}\right)^{n}$. Then $\lim _{n \rightarrow \infty} x_{n}=$
4. 1953.

Notice that, $2013=3 \times 11 \times 61$.
Therefore, when $1 \leqslant k \leqslant 60$,
$$
\frac{k!}{2013} \notin \mathbf{Z}, \cos \frac{k!\pi}{2013} \in(-1,1).
$$

Hence $\lim _{n \rightarrow \infty} \cos ^{n} \frac{k!\pi}{2013}=0$.
And when $k \geqslant 61$, $\frac{k!}{2013}$ is an integer, and always an even number, at this time, $\cos \frac{k!\pi}{2013}=1$, accordingly, $\lim _{n \rightarrow \infty} \cos ^{n} \frac{k!\pi}{2013}=1$.
$$
\text { Therefore, } \lim _{n \rightarrow \infty} x_{n}=2013-60=1953 \text {. }
$$