On the divisor function and class numbers of real quadratic fields. II
type: Item
instance of: scholarly article
title: On the divisor function and class numbers of real quadratic fields. II
zbMATH Open document ID: 0723.11052
author: Richard A. Mollin
published in: Proceedings of the Japan Academy. Series A
publication date: 1990-01-01T00:00:00Z
review text: This paper is a continuation of the author's preceding one [Proc. Japan Acad., Ser. A 66, No. 5, 109--111 (1990; Zbl 0714.11068)] in which he provided lower bounds for the class number of real quadratic fields of narrow Richaud-Degert type in terms of the divisor function \(\tau(x)\). In this paper, he proves the following for the class number \(h(d)\) of real quadratic fields \(\mathbb Q(\sqrt{d})\) \((d=\ell^ 2+r)\) of E.R.D. type: Let \[ P=\{\text{primes }p_ i:\;p_ i\mid \ell \text{ with } (r,p_ i)=1,\quad r\not\equiv 1\pmod{p_ i}\} \] and put \[ P(\Delta)=\left\{s=\prod_{i}p_ i^{e_ i}: e_ i\geq 1, s\leq \sqrt{\Delta /2}\right\} \] (\(\Delta\) : discriminant of \(\mathbb Q(\sqrt{d}))\). Then \(h(d)\geq \tau(q)\) for all \(q\in P(\Delta)\).
reviewed by: Hideo Yokoi
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 4191812
zbMATH Keywords: lower bounds
zbMATH Keywords: class number
zbMATH Keywords: real quadratic fields
zbMATH Keywords: narrow Richaud-Degert type
zbMATH Keywords: divisor function
MaRDI profile type: Publication
full work available at URL: pjaa.66.274
OpenAlex ID: W4255990954
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DOI: 10.3792/PJAA.66.274
description: scientific article; zbMATH DE number 4191812