Independence of Artin L-functions

Abstract

AbstractLet {K/\mathbb{Q}} be a finite Galois extension. Let {\chi_{1},\ldots,\chi_{r}} be {r\geq 1} distinct characters of the Galois group with the associated Artin L-functions {L(s,\chi_{1}),\ldots,L(s,\chi_{r})}. Let {m\geq 0}. We prove that the derivatives {L^{(k)}(s,\chi_{j})}, {1\leq j\leq r}, {0\leq k\leq 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}.