I know the Brun-Titchmarsh theorem states the following.
For x > q, $$\pi(x;q,a) \leq \frac{2}{1-\theta}\frac{x}{\phi(x)\log{x}}$$
where $\pi(x;q,a)$ denotes the set of primes less than x congruent to a modulus q, $\theta = \frac{\log{q}}{\log{x}} $
Is there a known bound on the quantity $$|\pi(x;q,a) - \frac{Li(x)}{\phi(x)}| $$ that holds uniformly for $q < x$.