Primes in arithmetic progression with a moduli equal to a power of 2

Question

I am currently looking for a result stronger than Siegel-Walfisz theorem, which gives an upper bound on the error term $|\pi(x,a,b)-\frac{\pi(x)}{\phi(a)}|$ for particular $a$.

The Siegel Walfisz is true for $a<(\ln x)^A$. Relying on this work : https://eudml.org/doc/207492, I conclude that I can assume that $|\pi(x,a,b)-\frac{\pi(x)}{\phi(a)}| < \frac{Cx}{\phi(a)(\ln x)^A}$ uniformly for $a = p^k$ ($p\geq 3$) and $a^{8/3+\epsilon} < x$ (correct me if I misunderstand something).

Regarding this recent work, http://link.springer.com/article/10.1007/s11139-006-0250-4, can I assume the same thing for $a=2^k$ and $a^3<x$ ? What is so special about $p=2$ ? Is there a reference on this case ?

Answer 1

There is nothing special about $p=2$; just that there are some technical differences, e.g. the group of reduced residues looks a little different than for odd prime powers. In the case of powers of a fixed prime, one has a stronger Siegel-Walfisz theorem because the characters can essentially be thought of as polynomials and Vinogradov's method (and Vinogradov's zero free region) applies. A generalized version of this was worked out in an early paper of Iwaniec On zeros of Dirichlet's L-series in Inventiones, 1974 (following the work of Postnikov, and Gallagher), and he does allow for $q$ to be a power of $2$. He gives a good zero free region for integers $q$ for which the radical of $q$ (namely $\prod_{p|q} p$) is small, with the exception of a possible Siegel zeros for a real character, see Theorem 2. Of course for powers of $2$ (or any fixed prime), there are only primitive real characters $\pmod 4$ or $\pmod 8$ and so the Siegel zero question is moot. Once a Vinogradov type zero free region is at hand, one can combine this with the usual zero density results and then obtain the desired stronger Siegel-Walfisz results.