On the zeta functions attached to ray classes
type: Item
instance of: scholarly article
title: On the zeta functions attached to ray classes
zbMATH Open document ID: 0941.11046
author: Wolfgang Jenkner
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-03-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/225598
full work available at URL: jtnb7-1.html
review text: Let \(K/\mathbb{Q}\) be a finite extension (the author is concerned mainly with quadratic extensions) and let \({\mathfrak m}(\neq(0))\) be an ideal of the ring of integers of \(K\), \({\mathcal O}(K)\). Denote by \(I({\mathfrak m})\) the group generated by the ideals \({\mathfrak a}\) of \({\mathcal O}(K)\) prime to \({\mathfrak m}\) and by \(P({\mathfrak m})\) the subgroup of principal ideals. Let \[ P_{\mathfrak m}= \bigl\{(x) \in P({\mathfrak m}) \mid x\equiv 1\mod^\times {\mathfrak m} \bigr\} \] and \[ P_{{\mathfrak m}^+}= \bigl\{(x)\in P_{\mathfrak m} \mid\sigma_1 (x)>0,\dots, \sigma_r(n)> 0\bigr \}, \] where \(\sigma_1, \dots,\sigma_r\) are the \(\mathbb{Q}\)-monomorphisms \(K\to \mathbb{R}\). For a group \(\ell\) such that \(P_{{\mathfrak m}^+}\leq\ell\leq P({\mathfrak m})\) let \(H(\ell)= I({\mathfrak m})/ \ell\). For \(\text{Re} s>1\) the zeta function attached to the ray class \({\mathfrak a}\ell\) is defined by \[ \zeta(s,{\mathfrak a}\ell)= \sum_{n \ell ={\mathfrak a}\ell} {n{\mathfrak m}^s \over N{\mathfrak n}^s} \] where \(N\) denotes the norm. The main results of this paper are related to the problem of obtaining inequalities for \[ \int\bigl|\zeta(s,{\mathfrak a} \ell)\bigr |^2dt \] in the region \({1\over 2}\leq \sigma<1\), \(s=\sigma +it\). After proving some elementary results suggested by \textit{K. Prachar}'s [Primzahlverteilung. Springer-Verlag (1978; Zbl 0394.10001)] and after obtaining the functional equation and approximate functional equation (using ideas from \textit{K. Chandrasekharan} and \textit{R. Narasimhan} [Math. Ann. 152, 30-64 (1963; Zbl 0116.27001)], the author's main result is as follows. Consider the quadratic case and let \[ \zeta^*(s,{\mathfrak a}\ell) = \sum_{n \geq N{\mathfrak m}} {a({\mathfrak m}, {\mathfrak a},n)\over N {\mathfrak m}^s},\quad Z^* (\sigma)= N{\mathfrak m}^{2\sigma} \sum_{n\geq N{\mathfrak m}} {a({\mathfrak m}, {\mathfrak a}, n)^2 \over N{\mathfrak n}^{2\sigma}}, \] where \(a({\mathfrak m},{\mathfrak a}, n…
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 858010
zbMATH Keywords: zeta functions of algebraic number fields
zbMATH Keywords: ray-classes
zbMATH Keywords: quadratic fields
MaRDI profile type: Publication
OpenAlex ID: W2314793113
cites work: The approximate functional equation for a class of zeta-functions
cites work: Q3735800
cites work: Topics in multiplicative number theory
cites work: Q4175321
cites work: Q5523060
cites work: Q3909097
DOI: 10.5802/JTNB.126
description: scientific article; zbMATH DE number 858010