Class group \(L\)-functions
type: Item
instance of: scholarly article
title: Class group \(L\)-functions
zbMATH Open document ID: 0838.11058
DOI: 10.1215/S0012-7094-95-07901-0
published in: Duke Mathematical Journal
publication date: 1996-06-03T00:00:00Z
review text: Let \(K = \mathbb{Q}(\sqrt{-D})\) be an imaginary quadratic field with class number \(h\), and let \(\chi\) be any one of the \(h\) characters of the class group of \(K\). Then the class group \(L\)-functions \(L_K (s,\chi)\) are defined as natural generalizations of the Dedekind zeta-function. The authors consider various mean values of these \(L\)-functions on the critical line, but their ultimate goal is estimating a single \(L\)-function in the \(D\)-aspect. A ``trivial'' estimate is \(\ll D^{1/4} \log^2D\) for fixed \(s = 1/2 + it\) (the implied constant may depend on \(s)\); this follows from the functional equation by the convexity principle. If \(\chi\) is a real character, then \(L_K(s,\chi)\) is a product of two Dirichlet \(L\)-functions, for which well known bounds of Burgess are available. However, the case of complex characters is much more problematic. The average of \(L_K (s,\chi)\) over \(\chi\) reduces to Riemann's zeta-function very precisely. A deeper problem is the mean square, and here a saving by \(D^{-1/28+\varepsilon}\) in the error term is obtained. However, this does not imply any nontrivial estimate for a single \(L\)-function. For this purpose, an ``amplifier'' is attached to the terms of the mean-square; it is a factor which emphasizes the function to be estimated. This idea has been applied by the authors previously to character sums and automorphic \(L\)-functions. The result is \(L_K (s,\chi) \ll D^{1/4-\alpha+\varepsilon}\) with \(\alpha = 1/1156\), but this bound is conditional: three alternative sufficient conditions are given in terms of the class number \(h\), the nontriviality of short field character sums, or the existence of sufficiently many small primes with \(({- D \over p}) = 1\). For instance, the last mentioned property fails for at most finitely many \(D\) in intervals of the type \((X, X^2)\). Also, the estimate holds unconditionally if all the prime factors of \(D\) are bounded by \(D^{\alpha^2}\). The proof is compli…
reviewed by: Matti Jutila
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 809306
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: estimation of single \(L\)-functions
zbMATH Keywords: class group \(L\)-functions
zbMATH Keywords: amplifier
zbMATH Keywords: class number
zbMATH Keywords: mean values
zbMATH Keywords: automorphic functions
zbMATH Keywords: spectral theory
author: W. Duke
author: J. B. Friedlander
author: H. Iwaniec
MaRDI profile type: Publication
cites work: On the least integers represented by the genera of binary quadratic forms
cites work: On Character Sums and Primitive Roots†
cites work: Class Groups of Quadratic Fields
cites work: Kloosterman sums and Fourier coefficients of cusp forms
cites work: Hyperbolic distribution problems and half-integral weight Maass forms
cites work: Bounds for automorphic \(L\)-functions
cites work: Bounds for automorphic \(L\)-functions. II
cites work: A mean-value theorem for character sums
cites work: Q3710578
cites work: Extension of conjectures of Cohen and Lenstra
cites work: Q3671491
cites work: Q5750921
cites work: Q4052222
cites work: On a paper of Baker and Schinzel
cites work: Q4846158
cites work: \(L^ \infty\) norms of eigenfunctions of arithmetic surfaces
cites work: Heegner points, cycles and Maass forms
cites work: Über die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik. (On the spatial distribution of the points in lattices with indefinite metric)
cites work: Q5798194
cites work: Q4141905
cites work: Über die Classenzahl quadratischer Zahlkörper
cites work: Q4405225
full work available at URL: s0012-7094-95-07901-0
OpenAlex ID: W1994752010
description: scientific article; zbMATH DE number 809306