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Explicit upper bounds of the residue at the point 1 of zeta functions of certain number fields. (Majorations explicites du résidu au point 1 des fonctions zêta de certains corps de nombres.) (French) Zbl 1040.11081

Author’s summary: We give good upper bounds on the quotient \((\zeta_K/ \zeta_k)(1)\) of the residues at \(s=1\) of the Dedekind zeta-functions of \(K\) and \(k\), provided that \(K/k\) is an abelian extension which is unramified at all the infinite places. To this end, when \(\chi\neq 1\) is any Artin character of such an unramified at all the infinite places abelian extension \(K/k\), we give upper bounds on the modulus of \(L(1, \chi,K/k)\) that depend on the residue at \(s=1\) of the Dedekind zeta function of \(k\) and of the norm of the conductor of \(\chi\). These bounds are generalizations of the ones we proved in the author’s paper [S. Louboutin, C. R. Acad. Sci., Paris, Sér. I 316, 11–14 (1993; Zbl 0774.11051)]. We have used these bounds in [S. Louboutin, R. Okazaki and M. Olivier [Trans. Am. Math. Soc. 349, 3657–3678 (1997; Zbl 0893.11045)] to determine all the normal but non-abelian CM-fields of degree 12 with relative class number one. We have also used them in [S. Louboutin and R. Okazaki, Proc. Lond. Math. Soc. (3) 76, 523–548 (1998; Zbl 0891.11054)] to determine all the dihedral CM-fields of 2-power degrees with relative class number one.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11R29 Class numbers, class groups, discriminants
11R21 Other number fields
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