A purely analytical lower bound for \(L(1,\chi )\)
type: Item
instance of: scholarly article
title: A purely analytical lower bound for \(L(1,\chi )\)
zbMATH Open document ID: 1256.11046
author: Olivier Ramaré
published in: Annales Mathématiques Blaise Pascal
publication date: 2009-11-11T00:00:00Z
full work available at URL: https://eudml.org/doc/10578
review text: It is well-known that genus theory of quadratic forms and the Dirichlet class number formula gives the bound \(L(1,\chi)\geq \frac{2^{\omega(q)-1}\pi}{\sqrt{q}}\), where \(\omega(q)\) denotes the number of distinct prime factors of \(q\) and \(\chi\) is an odd primitive quadratic Dirichlet character of conductor \(q\). In the paper under review the author gives a simple and purely analytical proof of a lower bound of similar strength, namely he proves that \[ L(1,\chi)\geq \frac{2^{\omega(q)-1}\pi}{7.5\sqrt{q}}. \] Instead of the theory of quadratic forms and the Dirichlet class number formula, the author uses the functional equation of \(\Phi(s)=(2\pi/\sqrt{q})^{-s}\Gamma(s)\zeta(s)L(s,\chi)\) and some basic properties of the Dirichlet convolution \(\mu^2\star \chi\), where \(\mu\) denotes the Möbius function.
reviewed by: Łukasz Pańkowski
Mathematics Subject Classification ID: 11M20
Mathematics Subject Classification ID: 11L05
zbMATH DE Number: 5629439
zbMATH Keywords: lower bound for \(L(1,\chi)\)
zbMATH Keywords: Kloosterman sum
MaRDI profile type: Publication
OpenAlex ID: W2022023889
cites work: Gauss’ class number problem for imaginary quadratic fields
cites work: Q3344283
DOI: 10.5802/AMBP.265
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description: scientific article; zbMATH DE number 5629439