The sum of the squared residuals $\sum e_i^2$ is always zero.

The sum of the residuals $\sum (y_i - \hat{y}_i)$ is always zero, provided an intercept $\beta_0$ is included in the model.

The sum of residuals is only zero if the correlation $r$ is equal to $1$ or $-1$.

The sum of residuals is minimized to a negative value by the least-squares method.