Independence of Artin L-functions

Abstract

Abstract Let K / ℚ {K/\mathbb{Q}} be a finite Galois extension. Let χ 1 , … , χ r {\chi_{1},\ldots,\chi_{r}} be r ≥ 1 {r\geq 1} distinct characters of the Galois group with the associated Artin L-functions L ⁢ ( s , χ 1 ) , … , L ⁢ ( s , χ r ) {L(s,\chi_{1}),\ldots,L(s,\chi_{r})} . Let m ≥ 0 {m\geq 0} . We prove that the derivatives L ( k ) ⁢ ( s , χ j ) {L^{(k)}(s,\chi_{j})} , 1 ≤ j ≤ r {1\leq j\leq r} , 0 ≤ k ≤ m {0\leq k\leq m} , are linearly independent over the field of meromorphic functions of order < 1 {<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 {<1} .