3. $\sum_{k=1}^{207}\left[\sqrt[4]{\frac{2007}{k}}\right]$ The value is $\qquad$
3.2167 .

The value sought is the number of lattice points within the region bounded by the curve $y=\sqrt[4]{\frac{2007}{x}}$ and $x>0$, $y>0$.
Changing the column-wise counting to row-wise counting, the value sought is
$$
\begin{array}{l}
\sum_{i=1}^{6}\left[\frac{2007}{i^{4}}\right]=2007+125+24+7+3+1 \\
=2167
\end{array}
$$