Upper bounds on L (1, χ ) taking into account ramified prime ideals

Stéphane R. Louboutin (1)

1 I2M - Institut de Mathématiques de Marseille

Let chi range over the non-trivial primitive characters associated with the abelian extensions L/K of a given number field K, i.e. over the non-trivial primitive characters on ray class groups of K. Let f(chi) be the norm of the finite part of the conductor of such a character. It is known that vertical bar L(1, chi)I <= 1/2-Res(s=1)(xi(K)(s)) log f(chi) + O(1), where the implied constants in this 0(1) are effective and depend on K only. The proof of this result suggests that one can expect better upper bounds by taking into account prime ideals of K dividing the conductor of chi, i.e. ramified prime ideals. This has already been done only in the case that K = Q. This paper is devoted to giving for the first time such improvements for any K. As a non-trivial example, we give fully explicit bounds when K is an imaginary quadratic number field.

Mots clés: L-function; Dedekind zeta function; Ray class group; Number field; Residues

Domaines: Théorie des nombres [math.NT]

Stéphane R. Louboutin. Upper bounds on L (1, χ ) taking into account ramified prime ideals. Journal of Number Theory, 2017, 177, pp.60 - 72. ⟨10.1016/j.jnt.2017.01.010⟩. ⟨hal-01581106⟩