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

  1. 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 $c_1$
  2. 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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    $\begingroup$ The first claim (Siegel-Walfisz) is only known under the additional hypothesis that $q \leq \log^A x$ for some fixed $A$ (with $c_1$ depending (ineffectively) on $A$). A more precise estimate is known if one admits the possibility of a term coming from an exceptional zero; see e.g. Davenport's "Multiplicative number theory". $\endgroup$ – Terry Tao Jan 16 '15 at 4:33
  • $\begingroup$ Ah yes, sorry. I sacrificed accuracy for brevity. I'll have a look at Davenport. $\endgroup$ – Stijn Jan 16 '15 at 4:35
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    $\begingroup$ An error term of roughly $x^\epsilon (x/q)^{1/2}$ for $q \leq x$ is predicted by the Cramer model, and was made explicitly by Montgomery (possibly with some log factor also), see ams.org/mathscinet-getitem?mr=427249 $\endgroup$ – Terry Tao Jan 16 '15 at 4:48
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    $\begingroup$ Under GRH, the error is just $O(\sqrt x\log x)$ (with explicit dependence of the constant on $q$), see e.g. Iwaniec and Kowalski. $\endgroup$ – Emil Jeřábek Jan 16 '15 at 12:30
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    $\begingroup$ See also page 426 of Montgomery-Vaughan "Multiplicative Number Theory I", particularly Conjecture 13.9. $\endgroup$ – Terry Tao Jan 16 '15 at 18:09
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To the last comment of Terry Tao: For a Vinograodov-Korobov estimate in APs (proved "elementarily" no less), see PRETENTIOUS MULTIPLICATIVE FUNCTIONS AND THE PRIME NUMBER THEOREM FOR ARITHMETIC PROGRESSIONS by DIMITRIS KOUKOULOPOULOS, especially Footnote 2.

http://dx.doi.org/10.1112/S0010437X12000802

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