Some remarks on Landau-Siegel zeros
type: Item
instance of: scholarly article
title: Some remarks on Landau-Siegel zeros
zbMATH Open document ID: 1008.11033
DOI: 10.1215/S0012-7094-02-11134-X
author: Alexandru Zaharescu
author: Peter Sarnak
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: The authors study the possible ``exceptional'' zero of a Dirichlet \(L\)-function \(L(s, \chi _D)\) for a real primitive character \(\chi _D \pmod{|D|}\) under certain hypotheses. Let \(E\) be an elliptic curve over \(\mathbb Q\). ``Hypothesis H'' assumes that all zeros of \(L(s, \chi _d)\) and \(L(s, E\otimes \chi _D)\) are either real or lie on the critical line. It is shown that this hypothesis (for Dirichlet \(L\)-functions only) implies the ineffective lower bound \(L(1, \chi _D) \geq c(\varepsilon)(\log |D|)^{-\varepsilon }\). Further, the full hypothesis H gives the effective lower bound \(L(1, \chi _D) \geq c(\eta)|D|^{-\eta }\) for any \(\eta >2/5\) if \(\chi _D(37)=-1\). Also, if \(L(s,E)\) has a zero of order \(m\) at \(s=1/2\), then the effective estimate \(L(1, \chi _D) \geq c(E, \varepsilon)|D|^{-(2+\varepsilon)/(m+1)}\) holds. The general scheme of the argument is to start from a suitable product of \(L\)-functions and to apply corresponding ``explicit formulae'' connecting arithmetic sums with zeros.
reviewed by: Matti Jutila
Mathematics Subject Classification ID: 11M20
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 1820880
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: Siegel zero
zbMATH Keywords: elliptic curves
MaRDI profile type: Publication
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full work available at URL: s0012-7094-02-11134-x
OpenAlex ID: W1998823764
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description: scientific article; zbMATH DE number 1820880