# Prime Number Theorem on APs under various conjectures

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

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

• 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". – Terry Tao Jan 16 '15 at 4:33
• Ah yes, sorry. I sacrificed accuracy for brevity. I'll have a look at Davenport. – Stijn Jan 16 '15 at 4:35
• 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 – Terry Tao Jan 16 '15 at 4:48
• 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. – Emil Jeřábek Jan 16 '15 at 12:30
• See also page 426 of Montgomery-Vaughan "Multiplicative Number Theory I", particularly Conjecture 13.9. – Terry Tao Jan 16 '15 at 18:09