Remark on upper bounds for \(L(1, \chi)\)
type: Item
instance of: scholarly article
title: Remark on upper bounds for \(L(1, \chi)\)
zbMATH Open document ID: 0867.11059
author: Shin-ichi Katayama
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-08-10T00:00:00Z
review text: Let \(p\) be an odd prime with \(p\equiv 1\pmod 4\). Further, let \(H(p)=\sqrt p(\log\sqrt p+1)/2\), \(W(p)=\sqrt p(\log\sqrt p+\gamma-1/2)/2\) and \(O(p)=\log(2(\begin{smallmatrix} n+\omega\\ n\end{smallmatrix})/\sqrt p)\), where \(n=(p-1)/4\), \(\omega=(1+\sqrt p)/2\) and \(\nu\) is Euler's constant. In this paper, the author proves that \[ \begin{aligned} O(p) &<W(p)< H(p)\qquad \text{if }5\leq p \leq 661\\ W(p) &<O(p)< H(p)\qquad \text{otherwise}.\end{aligned} \] The above inequalities are concerned with upper bounds for the value \(L(1,\chi)\) of Dirichlet \(L\)-functions.
Mathematics Subject Classification ID: 11M20
zbMATH DE Number: 938088
zbMATH Keywords: special values
zbMATH Keywords: inequalities
zbMATH Keywords: Dirichlet \(L\)-functions
reviewed by: Le Maohua
MaRDI profile type: Publication
full work available at URL: pjaa.72.89
OpenAlex ID: W2020178598
cites work: Q4477379
cites work: On the least solution of Pell’s equation
cites work: Q3941473
cites work: Q4695481
cites work: Q4732538
cites work: An upper bound for the period of the simple continued fraction for \(\sqrt D\)
DOI: 10.3792/PJAA.72.89
description: scientific article; zbMATH DE number 938088