On Artin's L-functions. I

Nicolae, Florin

doi:10.1515/crll.2001.073

Cited by 11 publications

(11 citation statements)

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“…For Artin L-functions a similar but weaker result has recently been proved by Nicolae [8]. In the remaining sections we will show suitable subsets of Artin L-functions satisfy the hypotheses of Theorem 2 so that the linear independence of their derivatives follows.…”

mentioning

confidence: 55%

General linear independence of a class of multiplicative functions

Molteni

2004

Arch. Math.

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mentioning

confidence: 55%

General linear independence of a class of multiplicative functions

Molteni

2004

Arch. Math.

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“…

This is the continuation of the research on Artin's L-functions initiated in [4]. The knowledge of [4] is not necessary in order to understand the result presented here.

Let a n , n f 1, be complex numbers.

…”

mentioning

confidence: 73%

On Artin’sL-functions. II: Dirichlet coefficients

Nicolae¹

2005

Journal für die reine und angewandte Mathematik (Crelles Journa

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This is the continuation of the research on Artin's L-functions initiated in [4]. The knowledge of [4] is not necessary in order to understand the result presented here.Let a n , n f 1, be complex numbers. If for a real number y the summatory function P nex a n is Oðx y Þ as x ! y, then the Dirichlet series P y n¼1 a n n s converges in the half-plane ReðsÞ > y. For a non-principal Dirichlet character modulo m it holds for every x f 1 P nex wðnÞ e jðmÞ;so P nex wðnÞ is Oð1Þ and the series P y n¼1 wðnÞ n s converges for ReðsÞ > 0. The main result of this paper is Theorem. Let K=Q be a finite Galois extension, let w be an r-dimensional character of the Galois group G ¼ GalðK=QÞ which does not contain the principal character, let L ur ðs; w; K=QÞ be the unramified part of the corresponding Artin L-function, and let L ur ðs; w; K=QÞ 1 r ¼ P y n¼1 a n n s for ReðsÞ > 1.(i) It holds ja n j e 1 for every n f 1.(ii) The summatory function P nex a n is oðxÞ as x ! y.The theorem says in the case K ¼ Qðe 2pi m Þ that P nex wðnÞ is oðxÞ as x ! y, which is weaker than P nex wðnÞ is Oð1Þ. It should be seen as an assertion about a general Artin L-function in the case of a non-abelian extension K=Q.Brought to you by | California Institute of Technology Authenticated Download Date | 5/27/15 12:45 AM

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“…We remark that Nicolae [6] has recently obtained a weaker result of this type in the special case of Artin L-functions. Such a class does in fact contain the class S as well as several important L-functions, in particular the Artin L-functions and the automorphic L-functions.…”

Section: Introductionmentioning

confidence: 81%

“…Therefore, (6) implies that each side is identically one, and Lemma 2 follows. In fact, in this case the zeros and poles of hðsÞ consist of finitely many ''vertical progressions'' i y j ð pÞ log p þ i 2p log p Z.…”

Section: Introductionmentioning

confidence: 92%

Linear independence of L-functions

2006

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We prove the linear independence of the members of a large class of L-functions defined axiomatically. Such a class, more general than the Selberg class, contains in particular the Artin and the automophic L-functions as well as their reciprocal and derivatives. Thus these results supersede all previous results on linear independence of L-functions. The main step in the proof is the multiplicity one property

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On Artin's L-functions. I

Cited by 11 publications

References 4 publications

General linear independence of a class of multiplicative functions

General linear independence of a class of multiplicative functions

On Artin’sL-functions. II: Dirichlet coefficients

Linear independence of L-functions

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