4. If the product of the first 2011 positive integers
$$
1 \times 2 \times \cdots \times 2011
$$

can be divided by $2010^{k}$, then the maximum value of the positive integer $k$ is
4.30.
$$
2010=2 \times 3 \times 5 \times 67 \text {. }
$$

In $1 \times 2 \times \cdots \times 2011$, the exponent of 67 is
$$
\left[\frac{2011}{67}\right]+\left[\frac{2011}{67^{2}}\right]+\cdots=30 \text {. }
$$

Obviously, the exponents of $2,3,5$ in $1 \times 2 \times \cdots \times 2011$ are all greater than 30.
Therefore, the maximum value of $k$ is 30.