Independence of Artin L-functions

Abstract

Let $K/\mathbb{Q}$ be a finite Galois extension. Let ${\chi }_1,\dots ,{\chi }_r$ be $r\ge 1$ distinct characters of the Galois group with the associated Artin L-functions $L(s,{\chi }_1),\dots ,L(s,{\chi }_r)$. Let $m\ge 0$. We prove that the derivatives $L^{(k)}(s,{\chi }_j)$, $1\le j\le r$, $0\le k\le m$, are linearly independent over the field of meromorphic functions of order $<1$. From this it follows that the L-functions corresponding to the irreducible characters are algebraically independent over the field of meromorphic functions of order $<1$.

10 pages, minor changes, reference added