Proof of convergence of $L'\left(1,\chi\right)$

Question

can someone give me a good reference for a clear proof of the convergence of $L'\left(1,\chi\right)$, $\chi$ real-valued, non-principal Dirichlet character?

Thanks in advance.

Answer 1

Dirichlet's test for convergence says if $a_1,a_2,\dots$ is a sequence of positive reals decreasing to zero, and $b_1,b_2,\dots$ is a sequence of complex numbers such that $\bigl|\sum_1^nb_k\bigr|$ is bounded independent of $n$, then $\sum_1^{\infty}a_nb_n$ converges.

Let $a_n=(1/n)\log n$ (strictly speaking, not decreasing, since $a_1=0$, but decreasing from some point on, which is good enough), let $b_n=\chi(n)$, then $b_n$ is periodic and sums to zero over a period, so the hypotheses of Dirichlet's test are met, so $\sum_1^{\infty}(1/n)\chi(n)\log n$ converges.

I don't know this for certain, but I suspect this is the problem that led Dirichlet to devise his test.