24. Let $\chi$ be a non-principal character modulo $k$. Prove:
(i) When $s>0$, the series $\sum_{n=1}^{\infty} \chi(n) n^{-s}$ converges, and when $s>1$,
$$L(s, \chi)=\sum_{n=1}^{\infty} \chi(n) n^{-s}=\prod_{p}\left(1-\frac{\chi(p)}{p^{s}}\right)^{-1}$$
(ii) When $s>1$,
$$L^{-1}(s, \chi)=\sum_{n=1}^{\infty} \mu(n) \chi(n) n^{-s}$$
(iii) When $s>1$,
$$\ln L(s, \chi)=\sum_{n=2}^{\infty} \Lambda(n) \chi(n)(\ln n)^{-1} n^{-s} ;$$
(iv) When $s>1$,
$$-\frac{L^{\prime}(s, \chi)}{L(s, \chi)}=\sum_{n=1}^{\infty} \Lambda(n) \chi(n) n^{-s}$$