jmc

Let $a=2003/2004$. The given equation is equivalent to $$a{x}^2+x+1=0.$$ If the roots of this equation are denoted $r$ and $s$, then $$rs=\frac{1}{a}\text{and}r+s=-\frac{1}{a},$$ so $$\frac{1}{r}+\frac{1}{s}=\frac{r+s}{rs}=\boxed{-1}.$$