= Nonvanishing of a nonprincipal Dirichlet L-function at one
= Nonvanishing of nonprincipal Dirichlet L-functions at one
{synonym}
For every nonprincipal <Dirichlet character>, $L(1,\chi)\ne0$. A nonreal character is covered by <Euler product positivity for L-function nonvanishing> at height zero, since its squared character is nonprincipal.
For a real character, suppose $L(1,\chi)=0$. The simple zeta pole would cancel, making $F(s)=\zeta(s)L(s,\chi)$ an <entire function>. Its <nonnegative zeta-times-real-L coefficients> satisfy $a_n\ge0$ and $a_{m^2}\ge1$. Termwise derivatives at two give $(-1)^kF^{(k)}(2)=\sum_na_n(\log n)^kn^{-2}$. The <Taylor series> of this entire function at two converges at zero. Every term there is nonnegative, and <Tonelli theorem> identifies its sum as
$$
F(0)=\sum_{k\ge0}\frac{2^k}{k!}\sum_n\frac{a_n(\log n)^k}{n^2}=\sum_na_n.
$$
The last series diverges because all square coefficients are at least one. This contradiction proves the claim. It is the positive-coefficient mechanism behind the more general <Landau theorem for a Dirichlet series with nonnegative coefficients>.
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