The Siegel-Walfisz theorem

When considering primes in arithmetic progressions, one is naturally led to study the zeros of Dirichlet $L$-functions. As will be seen, one can prove the prime number theorem in arithmetic progressions, that is $\pi(x;a,q)=\frac{\text{Li}(x)}{\varphi(q)}+O_q(x\exp(-c_q\sqrt{\log x}))$, using arguments very analogous to proving the prime number theorem with the help of the zero free region of the Riemann zeta function $\zeta(s)$. However, things get much more complicated when uniformity in the modulus $q$ is required. Such uniformity was for example required in this post on the ternary Goldbach problem. The reason for increased complexity is that the location of the zeros of $L(s,\chi)$ may depend intricately on the modulus $q$ of $\chi$, as well as $\chi$ itself. For the $\zeta$ function no such problem exists, and given any finite rectangle in the plane, one can determine whether it contains zeros of $\zeta(s)$ just by calculating an integral arising from the principle of argument. For general $L$-functions, zeros lying too close to the line $\sigma=1$ could ruin all hope of uniformity in the prime number theorem for arithmetic progressions.

Fortunately, there is a classical theorem , which states that zeros of $L(s,\chi)$ close to the line $\sigma=1$, say with $\sigma>1-\frac{c}{\log q}$, with $c$ a suitable constant, must be exceptional in three different ways. First, $\chi$ must be a non-principal real primitive character; second, $s$ must be real, and third such a zero is at most unique. We shall prove this result and deduce Siegel's theorem, stating that $L(s,\chi)$ has no zeros with $s>1-C(\varepsilon)q^{-\varepsilon}$ for any fixed $\varepsilon>0$. This in turn implies the Siegel-Walfisz theorem, which gives a uniform error term from primes in arithmetic progressions up to $q\leq (\log x)^{M}$ for any fixed $M$. A defect of these theorems is that the constant $C(\varepsilon)$ is completely ineffective; the proof gives no bounds for it. Nevertheless, 80 years after Siegel's discovery, the result is essentially the best known. Already showing that the single real zero with $s>1-\frac{c}{\log q}$, known as the Landau-Siegel zero, never exists, would have profound consequences, including an improvement of the Brun-Titchmarsh theorem (see this post), an improved Siegel-Walfisz theorem, and better estimates for the class numbers of imaginary quadratic fields. We follow Davenport's Multiplicative Number Theory.