Independence of Artin L-functions

Abstract

Let {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}.

Two semigroup rings associated to a finite set of germs of meromorphic functions

“…. , f r are algebraically independent over C. This result was extended in [5] to the field M <1 of meromorphic functions of order < 1. Let F be a field such that C ⊂ F ⊂ M <1 .…”

On the semigroup ring of holomorphic Artin L-functions

“…Using the notations from the Introduction, from [10, Theorem 1] and [5,Corollary 9], it follows that the map…”

“…. , f r are algebraically independent over C. This result was extended in [5] to the field M <1 of meromorphic functions of order < 1.…”

“…Let F be a field such that C ⊆ F ⊆ M <1 . From [10, Theorem 1] and [5,Corollary 9], we have the F-algebra isomorphism F[Hol(s 0 )] ∼ = F[H(s 0 )], therefore, the Artin's conjecture at s 0 can be stated as…”

On supercharacter theoretic generalizations of monomial groups and Artin's conjecture

“…. , f r are algebraically independent over C, a result extended later in [6], where it was proved that f 1 , . .…”