Comparing L s with its truncated Euler product and generalization

Description

Comparing L(s, ?) with its truncated Euler product and generalization O. Ramare April 17, 2009 Abstract We show that any L-function attached to a non-exceptionnal Hecke Grossencharakter ? may be approximated by a truncated Euler prod- uct when s lies near the line 0 such that no L-function L(s,?) has a zero ? in the region

• dirichlet characters


• euler prod- uct when


• no real


• let ?


• euler product


• see also


• exceptional characters


• exceptional characters can


• hecke grossencharakter

Sujets

• Dirichlet character
• Euler product
• See also

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 37
• Langue: English

Extrait

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ComparingL(s, χ) with its truncated Euler product and generalization

O.Ramar´e

April 17, 2009

Abstract

We show that anyL-function attached to a non-exceptionnal Hecke Grossencharakter Ξ may be approximated by a truncated Euler prod- uct whenslies near the line<sThis leads to some refined= 1. bounds onL(s,Ξ).

Introduction and results

ForL(1, χ), see [8], [16] and [1]. We first need to fix some terminology. We select a number fieldK/Qbe a number field of degreedand discriminant Δ. We denote its norm by N, as a shortcut to N K/Q shall consider Hecke Grossencharakters Ξ with (finite). We idealf, of normq, and associated with some finite set of infinite places. The conductorfbeing fixed, the main Theorem of [5] tells us there exists an absolute constantC >0 such that noL-functionL(s,Ξ) has a zeroρin the region C <ρ≥1−(1) Log max(qΔ, qΔ|=s|) except at most one such character; this potential exception is real valued and may have at most one real zeroβin this region. We refer to this hypothetical character as the exceptional character and term the remaining ones as being non-exceptional. See also [11]. In the case of Dirichlet characters, i.e.K=Q, we know from [13] that we may takeC= 1/6.3958.

AMS Classification. Primary : 11R42, 11M06 ; Secondary : 11M20 Keywords: Hecke Grossencharakter, DirichletL-functions

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