On Artin's L-funcions. II: Dirichlet Coefficients

Abstract

Let $K/\Q$ be a finite Galois extension, let $\chi$ be a character of the Galois group $G=\Gal(K/\Q)$ which does not contain the principal character, let $L_{ur}(s,\chi,K/\Q)$ be the unramified part of the corresponding Artin $L$-function, and let $$L_{ur}(s,\chi,K/\Q)^\frac{1}{\chi(1)}=\sum_{n=1}^\infty\frac{a_n}{n^s}$$ for $\Re(s)>1$. Then:\\ (i) The coefficients $a_n$ are algebraic numbers of the field $\Q(e^\frac{2\pi i}{|G|})$ and $|a_n|\leq 1$ for every $n\geq 1$ ;\\ (ii) The summatory function $\sum_{n\leq x}a_n$ is ${\bf o}(x)$ as $x\rightarrow\infty$.