Low-lying zeros of dihedral \(L\)-functions
type: Item
instance of: scholarly article
title: Low-lying zeros of dihedral \(L\)-functions
zbMATH Open document ID: 1028.11055
DOI: 10.1215/S0012-7094-03-11621-X
published in: Duke Mathematical Journal
publication date: 2003-06-23T00:00:00Z
review text: Let \(D>3\) be squarefree, \(D \equiv 3 \pmod{4}\), \(\psi \) a character of the ideal class group of the imaginary quadratic field \(\mathbb Q (\sqrt{-D})\), and consider the zeros near \(s=1/2\) of the \(L\)-functions \(L(s,\psi),\) assuming the Riemann hypothesis for such functions. There is a philosophy, due to N. Katz and P. Sarnak, predicting the distribution of low-lying zeros of various families of \(L\)-functions, and accordingly a ``Density Conjecture'' is formulated in the present case. This conjecture involves a test function of compact support, and the goal is to prove it for test functions with support as wide as possible. Results in this direction are obtained, summing either over all character for fixed \(D\), or averaging even over \(D\). Arithmetically, the Density Conjecture appears to be equivalent to an ``Euler Primes Conjecture'' concerning primes \(p\) such that \(4p = m^2+Dn^2\). Sieve methods are applied to count such primes.
reviewed by: Matti Jutila
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 1911546
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: zeros
zbMATH Keywords: quadratic fields
Wikidata QID: Q101167833
author: Étienne Fouvry
author: H. Iwaniec
MaRDI profile type: Publication
cites work: Q3996190
cites work: Q4842213
cites work: Q4275389
cites work: Low lying zeros of families of \(L\)-functions
cites work: Q4226082
cites work: Zeroes of zeta functions and symmetry
cites work: Small zeros of \(L\)-functions of modular forms
full work available at URL: s0012-7094-03-11621-x
OpenAlex ID: W1971470118
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description: scientific article; zbMATH DE number 1911546