It is known that if $(a, q)$ and $q\le (\ln x)^N$, then the following is true
$$\sum_{k\le x, k\equiv a\pmod{q}}\Lambda(k) = \frac{x}{\phi(q)} + O(x\exp(-C\sqrt{\ln x}))$$
where $C$ depends only on $N$. This error term appears because of the possibility of a counterexample to the Generalised Riemann Hypothesis. What would the error term be if the GRH was true?
Corollary 13.8 of Montgomery and Vaughn's Multiplicative Number Theory states that under the Generalized Riemann Hypothesis for the Dirichlet $L$ functions of characters $\chi$ modulo $q$, we have that $$\psi(x;q,a)=\sum_{\begin{array}{c} n\leq x\\ n\equiv a\ (q) \end{array}}\Lambda(n)=\frac{x}{\phi(q)}+O\left(x^{1/2}\log^{2}x\right).$$ In particular, if GRH is true, then we have the asymptotic $\psi(x;q,a)\sim\frac{x}{\phi(q)}$ for all $q\leq x^{1/2}/\log^3(x)$.