On the use of explicit bounds on residues of Dedekind zeta functions taking into account the behavior of small primes

Louboutin, Stéphane

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 559-573.

Nous donnons des majorants explicites des résidus au point $s=1$ des fonctions zêta ${\zeta }_K\left(s\right)$ des corps de nombres tenant compte du comportement des petits nombres premiers dans $K$. Dans le cas où $K$ est abélien, de telles majorations sont déduites de majorations de $\left|L\right(1,\chi \left)\right|$ tenant compte du comportement de $\chi$ sur les petits nombres premiers, pour $\chi$ un caractère de Dirichlet primitif. De nombreuses applications sont données pour illustrer l’utilité de tels majorants.

Lately, explicit upper bounds on $\left|L\right(1,\chi \left)\right|$ (for primitive Dirichlet characters $\chi$) taking into account the behaviors of $\chi$ on a given finite set of primes have been obtained. This yields explicit upper bounds on residues of Dedekind zeta functions of abelian number fields taking into account the behavior of small primes, and it as been explained how such bounds yield improvements on lower bounds of relative class numbers of CM-fields whose maximal totally real subfields are abelian. We present here some other applications of such bounds together with new bounds for non-abelian number fields.

Mots clés : $L$-functions, Dedekind zeta functions, number fields, class number.

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