# Siegel theorem

For any $\epsilon > 0$ there exists a $c = c(\epsilon) > 0$ such that for any non-principal real Dirichlet character $\chi$ of modulus $k$, $$L(1,\chi) > \frac{c(\epsilon)}{k^\epsilon} \ .$$ First proved by C.L. Siegel . An equivalent formulation is concerned with real zeros of L-functions: For any $\epsilon > 0$ there exists a $c_1 = c_1(\epsilon) > 0$ such that $L(z,\chi) > c_1(\epsilon) \neq 0$ for $z > 1 - c_1/k^\epsilon$ for any non-principal real Dirichlet character $\chi$. The constants $c(\epsilon)$ and $c_1(\epsilon)$ are non-effective, in the sense that for no $0 < \epsilon < 1/2$ there is a way to estimate them from below. For this reason, applications of Siegel's theorem have a non-constructive character. For example, if $h(-D)$ is the divisor class number of a quadratic field of discriminant $-D$, it follows from the theorem that $$h(-D) > c_2(\epsilon) D^{1/2-\epsilon}$$ with a non-effective constant $c_2(\epsilon)$ for $\epsilon < 1/2$. Similarly, the following estimate, which is uniform for $1 \le k \le \log x$, $(k,l) = 1$, $$\pi(x;k,l) - \frac{1}{\phi(k)} \int_2^x \frac{dt}{\log t} = O\left({ x e^{-c_3 \sqrt{\log x}} }\right)$$ where $\pi(x;k,l)$ is the number of primes $< x$ representable as $kn + l$, involves a non-effective constant $c_3$ (cf. also Distribution of prime numbers).