Let $\chi$ be a quadratic character with conductor $m$. Then $L(s,\chi) \zeta(s) = \zeta_K(s)$ is the Dedekind zeta function of the quadratic number field $K$ whose decomposition law is determined by $\chi$ (we have $K = {\mathbb Q}(\sqrt{m})$Q}(\sqrt{ \pm m})$for a suitable choice of the sign), hence$\zeta_K(s) = \sum c_n n^{-s}$, where$c_n$denotes the number of ideals with norm$n$. In particular,$c_n \ge 0$and$c_{n^2} \ge 1$since$(n)$has norm$n$. If we had$L(1,\chi) = 0$, this zero would cancel the pole of$\zeta(s)$, and$\zeta_K(s)$would be analytic for all$s$with real part$> 0$; on the other hand, the divergence of the harmonic series implies that$\zeta_K(s)$diverges at$s = \frac12$. Now we can strip off the background by observing that$c_n = \sum_{d \mid n} \chi(d)$. We can show directly that$c_n \ge 0$and$c_{n^2} \ge 1$and show that$\sum c_n n^{-1/2}$diverges. A slightly technical but elementary manipulation shows that $$\sigma = \sum_{t = 1}^{n^2} \frac{c_t}{\sqrt{t}} = 2n L(1,\chi) + O(1).$$ If we had$L(1,\chi) = 0$, this sum would be bounded: contradiction. With slightly more effort, but still completely elementary means, we can prove that $$\log n < \sigma < 2n L(1,\chi) + 6\phi(m) + 1,$$ from which we can easily deduce that $$L(1,\chi) > \frac12 e^{-6 \phi(m) - 2}.$$ This is a lousy lower bound, but it gives us what we want. The basic idea behind this proof is due to Mertens in the late 1890s. It was the first proof of Dirichlet's theorem that did not use the quadratic reciprocity law (Dirichlet showed that quadratic characters$\chi$could be written in the form$\chi(a) = (\frac{m}{a})$, and then showed that$L(1,\chi)$is a product of nonzero terms involving the class number of the quadratic number field$K$), and hence it was possible to use Dirichlet's theorem for closing the gap in Legendre's proof (the fact that this had already achieved by Kummer had not been noticed by then). Mertens [Wiener Berichte 106 (1897), 254-286; J. Reine Angew. Math. 117 (1897), 169-184] did not start with the Dedekind zeta function but rather with Dirichlet's proof that $$\sum_{n=1}^N d(n) = N \log N + (2\gamma - 1) + O(\sqrt{N} ,$$ where$d(n)$denotes the sum of divisors of$n$. Dirichlet proved this result about the same time he proved his theorem on primes in arithmetic progression; he admitted that his first proof of the nonvanishing of his L-series was complicated, so he dropped it altogether and replaced it by his proof of the class number formula. It is not unlikely that his lost first proof of the nonvanishing contained elements of Mertens's idea. Pieces of this story occur in Hasse's Vorlesungen über Zahlentheorie, Siegel's lectures in anlytic number theory (in German; have these been translated into English?) and Zagier's book on quadratic fields and forms (also in German). 1 The proof I am about to give has already been given; here's "the rest of the story". Let$\chi$be a quadratic character with conductor$m$. Then$L(s,\chi) \zeta(s) = \zeta_K(s)$is the Dedekind zeta function of$K = {\mathbb Q}(\sqrt{m})$, hence$\zeta_K(s) = \sum c_n n^{-s}$, where$c_n$denotes the number of ideals with norm$n$. In particular,$c_n \ge 0$and$c_{n^2} \ge 1$since$(n)$has norm$n$. If we had$L(1,\chi) = 0$, this zero would cancel the pole of$\zeta(s)$, and$\zeta_K(s)$would be analytic for all$s$with real part$> 0$; on the other hand, the divergence of the harmonic series implies that$\zeta_K(s)$diverges at$s = \frac12$. Now we can strip off the background by observing that$c_n = \sum_{d \mid n} \chi(d)$. We can show directly that$c_n \ge 0$and$c_{n^2} \ge 1$and show that$\sum c_n n^{-1/2}$diverges. A slightly technical but elementary manipulation shows that $$\sigma = \sum_{t = 1}^{n^2} \frac{c_t}{\sqrt{t}} = 2n L(1,\chi) + O(1).$$ If we had$L(1,\chi) = 0$, this sum would be bounded: contradiction. With slightly more effort, but still completely elementary means, we can prove that $$\log n < \sigma < 2n L(1,\chi) + 6\phi(m) + 1,$$ from which we can easily deduce that $$L(1,\chi) > \frac12 e^{-6 \phi(m) - 2}.$$ This is a lousy lower bound, but it gives us what we want. The basic idea behind this proof is due to Mertens in the late 1890s. It was the first proof of Dirichlet's theorem that did not use the quadratic reciprocity law (Dirichlet showed that quadratic characters$\chi$could be written in the form$\chi(a) = (\frac{m}{a})$, and then showed that$L(1,\chi)$is a product of nonzero terms involving the class number of the quadratic number field$K$), and hence it was possible to use Dirichlet's theorem for closing the gap in Legendre's proof (the fact that this had already achieved by Kummer had not been noticed by then). Mertens did not start with the Dedekind zeta function but rather with Dirichlet's proof that $$\sum_{n=1}^N d(n) = N \log N + (2\gamma - 1) + O(\sqrt{N} ,$$ where$d(n)$denotes the sum of divisors of$n\$. Dirichlet proved this result about the same time he proved his theorem on primes in arithmetic progression; he admitted that his first proof of the nonvanishing of his L-series was complicated, so he dropped it altogether and replaced it by his proof of the class number formula. It is not unlikely that his lost first proof of the nonvanishing contained elements of Mertens's idea.