A note on a deformation of Dirichlet's class number formula
type: Item
instance of: scholarly article
title: A note on a deformation of Dirichlet's class number formula
zbMATH Open document ID: 0762.11042
author: Shigeru Katayama
published in: Proceedings of the Japan Academy. Series A
publication date: 1993-01-17T00:00:00Z
review text: For a prime \(p\), let \(\mathbb{Q}(\sqrt p)\) be the real quadratic field. Then, \textit{T. Ono} gave a deformation of the classical Dirichlet class number formula in case \(p\equiv 1\pmod 4\) [Algebraic analysis, Pap. dedic. M. Sato, Vol. 2, 659-666 (1989; Zbl 0683.12005)] and the author gave it in case \(p\equiv 3\pmod 4\) [On a deformation of Dirichlet's class number formula, Res. Bull. Tokushima Bunri Univ. 42]. In this paper, for any real quadratic field \(\mathbb{Q}(\sqrt m)\) the author gives by using the same method the following deformation: for the class number \(h\), the fundamental unit \(\varepsilon\) and the discriminant \(d\) of \(\mathbb{Q}(\sqrt m)\), we put \(w=0\) if \(d\) has at least two distinct prime factors and \(w=1\) otherwise. Then \[ \sqrt{m^ w}\varepsilon^ h=2\bigl(\sum^{v-1}_{j=0}d_ j\bigr)+d_ v\quad(v=\varphi(d)/4), \] where \(d_ j\) are determined by the following recurrence relation: \(jd_ j=\sum^ j_{t=1}\alpha_ td_{j-t}\) \((d_ 0=1,\;1\leq j\leq v)\), \(\alpha_ t=-(\varphi(d)/\varphi(d/n)\mu(d/n)-\chi(t)\sqrt d)/2\), \(n=(t,d)\), \((1\leq t\leq v)\) \((\varphi\) denotes Euler's function, \(\chi\) a Kronecker character belonging to \(\mathbb{Q}(\sqrt m))\).
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 97929
zbMATH Keywords: real quadratic field
zbMATH Keywords: deformation
zbMATH Keywords: class number formula
reviewed by: Hideo Yokoi
MaRDI profile type: Publication
full work available at URL: pjaa.68.58
OpenAlex ID: W2071050572
cites work: Q4732538
DOI: 10.3792/PJAA.68.58
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description: scientific article; zbMATH DE number 97929