I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states:

- Unconditionally we have \begin{equation} \pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x \operatorname{exp}\left(-c_1 \sqrt{\log x}\right)\right) \end{equation} for some constant $c_1$
- Under GRH we have \begin{equation} \pi(x; a, q) = \frac{\operatorname{li}(x)}{\phi(q)} + O\left(x^{1/2 + \epsilon}\right) \end{equation} for all $\epsilon > 0$.

Do we know any further terms in these asymptotic expansions and are there any other conjectures which give better error terms?

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