Siegel–Walfisz theorem

Remarks

The proof follows the proof of the prime number theorem for each Dirichlet L-function $L(s, \chi)$ with $\chi$ modulo $q$. The classical zero-free region handles all zeros except a possible Siegel zero. Siegel's theorem says that for every $\varepsilon > 0$ such a real zero $\beta$ satisfies $\beta < 1 - c(\varepsilon) q^{-\varepsilon}$, and for $q \le (\log x)^N$ this is enough to make its contribution small. Siegel's theorem is ineffective, so $C_N$ cannot be computed from the proof. For example, with $N = 2$ and $q = 4$, the theorem gives $\psi(x; 4, 1) = x/2 + O(x e^{-c\sqrt{\log x}})$, a strong form of Dirichlet's theorem on primes in arithmetic progressions for primes $p \equiv 1 \pmod 4$. The range $q \le (\log x)^N$ is very short; for larger $q$ one has only average results such as the Bombieri–Vinogradov theorem.

Going deeper

For a student

The Siegel–Walfisz theorem is the standard input for the "small moduli" part of many arguments in analytic number theory. In proofs of the Bombieri–Vinogradov theorem and in the circle method for Vinogradov's theorem, the major arcs with small denominators are handled by it. Its constants are ineffective because of the possible Siegel zero; this is why results that rely on it, such as Vinogradov's theorem in its first form, give no explicit bound. Under the generalized Riemann hypothesis, the error improves to $O(x^{1/2} (\log x)^2)$ for all $q \le x$.