Artin’s L-functions: A Historical Approach

On Artin’s L-functions. Iii: One Dimensional Characters by Florin Nicolae at Berlin and Bucharest

Let K/Q be a finite Galois extension with the Galois group G, and let χ be a nontrivial irreducible character ofG. Artin’s conjecture predicts that the L-function L(s, χ,K/Q) is holomorphic in the whole complex plane ([1], P. 105). Let χ1, . . . , χr be the irreducible nontrivial characters of G. The corresponding L-functions L(s, χ1), . . . , L(s, χr) are algebraically independent over C ([2],...

On Holomorphic Artin L-functions

Let K/Q be a finite Galois extension, s0 ∈ C \ {1}, Hol(s0) the semigroup of Artin L-functions holomorphic at s0. We present criteria for Artin’s holomorphy conjecture in terms of the semigroup Hol(s0). We conjecture that Artin’s L-functions are holomorphic at s0 if and only if Hol(s0) is factorial. We prove this if s0 is a zero of an L-function associated to a linear character of the Galois gr...

Nearly Supersolvable Groups and Applications to Artin L-functions

In this note, we apply the group-theoretic method to study Artin’s conjecture, and introduce the notations of nearly nilpotent groups and nearly supersolvable groups to answer of a question of Arthur and Clozel. As an application, we show that Artin’s conjecture is valid for all nearly supersolvable Galois extensions of number fields as well as all solvable Frobenius extensions.

On Artin’s L-functions. II: Dirichlet Coefficients

On the history of Artin’s L-functions and conductors Seven letters from Artin to Hasse in the year 1930

In the year 1923 Emil Artin introduced his new L-functions belonging to Galois characters. But his theory was still incomplete in two respects. First, the theory depended on the validity of the General Reciprocity Law which Artin was unable at that time to prove in full generality. Secondly, in the explicit definition of L-functions the ramified primes could not be taken into account; hence tha...