Consider the function $$\vartheta(x;q,a)=\sum\_{p \leq x ,q|(p-a)}\log p=\frac{x}{\phi(q)}+O(\frac{x}{(\log x)^C}).$$
If the Riemann Hypothesis is ture, in the case $q=a=1$ we have $$|\psi(x)-x|<\frac{\sqrt{x}(\log x)^2}{8\pi}.$$
Now in the general case how well can the estimation be? (Can assume some further hypothesis like the GRH in the comment)



The best estimation of the function $\vartheta(x;q,a)$
It depends on what you want. On the one hand Montgomery conjectures that the estimate


$$\displaystyle \left \lvert \vartheta(x; q, a) - x \right \rvert = O\_\epsilon \left(q^{-1/2} x^{1/2 + \epsilon} \right),$$


holds under GRH for Dirichlet $L$-functions for $q \ll\_\epsilon x^{1 - \epsilon}$, and this error term is best possible for $q$ close to $x$ in size. On the other hand Iwaniec and Kowalski's book contains the following estimate from the same hypothesis:


$$\displaystyle \left \lvert \vartheta(x; q, a) - \frac{x}{\phi(q)} \right \rvert = O \left(x^{1/2} (\log qx) \right).$$


I am sure if one is careful one can track down an explicit implied constant, but I am not aware of this being done. Since most likely one cannot get away from dependence either on $q$ or some $\epsilon$, this may not be an interesting problem.


