Does zero sample correlation between regressor and residual mean Correlation between regressor and outcome = 0? In my econometrics professors' slides he states that:

”We said that $\sum_{i=1}^nX_i\hat{u}_i=0$ (where $\hat{u}$ are the residuals) implies zero sample correlation between X and Y“.

My problem is in understanding WHY this is (the statement in bold), or if he made a mistake and actually meant that each regressor always has zero correlation with residuals estimator, due to first order conditions used to grant OLS (even in a multivariate regression, not necessarily single variable regression).

IMPORTANT EDIT: X is the regressor in a single variable linear regression and Y is the outcome of such regression

That is clearly wrong as stated. As you say, orthogonality between regressors and residuals is true for any regression estimated by OLS. At the same time, the OLS estimator in a bivariate regression is known to be
$$
\hat\beta_1=\frac{\widehat{cov}(x,y)}{\widehat{var}(x)}
$$
Hence, if the statement were true, we would have a proof that, always, $\hat\beta_1=0$...
So I suppose that is a typo indeed.
