On the convergence of the Dirichlet series of an Artin L‐function

Abstract

AbstractLet \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$K/\mathbb {Q}$\end{document} be a finite Galois extension with the Galois group G, and let χ be a character of G with the associated Artin L‐function L(s, χ) defined in ℜ(s) > 1 by the Dirichlet series \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sum _{n=1}^\infty \frac{a_n}{n^s}$\end{document} with abscissa of convergence σc. Assume that L(s, χ) is holomorphic in the whole complex plane. If χ(1) = 1 then σc = 0, and if χ(1) > 1 then \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sigma _c\le \frac{\chi (1)}{2+\chi (1)}$\end{document}.