We consider the setting of online linear regression for arbitrary deterministic sequences, with the square loss. We are interested in regret bounds that hold uniformly over all vectors in $u $\in$ R^d$. Vovk (2001) showed a d ln T lower bound on this uniform regret. We exhibit forecasters with closed-form regret bounds that match this d ln T quantity. To the best of our knowledge, earlier works only provided closed-form regret bounds of 2d ln T + O(1).