Non-vanishing of Dirichlet $L$ function $L(s,\chi)$ for $Re s=1$

Question

Assuming $\zeta (1+it)\ne 0 , \forall t\in \mathbb R$ , is there an easy proof that $L(1+it , \chi ) \ne 0 , \forall t\in \mathbb R$ where $\chi$ is any non principal Dirichlet character ?

Answer 1

$$\log L(s,\chi) = \sum_{n=1}^\infty \frac{\Lambda(n)}{\log n} \chi(n) n^{-s}, \qquad \qquad\sum_{\chi \bmod q} \chi(n) =\varphi(q) 1_{n \equiv 1 \bmod q}$$ $$F(s) = \prod_{\chi \bmod q} L(s,\chi), \qquad\qquad \log F(s) = \sum_{\chi \bmod q} \log L(s,\chi) = \varphi(q)\sum_{n=1}^\infty \frac{\Lambda(n)}{\log n} 1_{n \equiv 1 \bmod q} n^{-s}$$ Once you proved $L(1,\chi)\ne 0$ for each non-principal $\chi \bmod q$ you get that $F(s)$ is analytic for $\Re(s) > 0$ except a simple pole at $s=1$.

Also $\log F(s)$ is a Dirichlet series with non-negative coefficients and it is analytic for $\Re(s) > 1$.

Thus the same argument $2(1+\cos \phi)^2 \ge 0$ as for $\zeta(s)$ applies to obtain $F(s) \ne 0$ for $\Re(s) \ge 1$

which implies $L(s,\chi) \ne 0$.

Finally $\frac{L'}{L}(s,\chi) = \mathcal{O}(\log^k s)$ for $\Re(s) > 1-\frac{A}{1+|\log |\Im(s)|}$ and Mellin inversion lets us prove the prime number theorem in arithmetic progessions.