A limit formula for \(\zeta (2k+1)\)
type: Item
instance of: scholarly article
title: A limit formula for \(\zeta (2k+1)\)
zbMATH Open document ID: 0949.11041
author: Yizhong Lan
published in: Journal of Number Theory
publication date: 2000-12-05T00:00:00Z
review text: The author studies the Riemann zeta-function \(\zeta(s)\) at positive odd integers \(s=n=2k+1\). Let \(q\) be a prime \(\equiv 1 \pmod n\) and let \(K\) be the subfield of degree \(n\) of the \(q\)-th cyclotomic field. Denote by \(I(K)\) the set of all non-zero integral ideals in \(K\) and by \(\mathcal O_{K,0}\) the \(\mathbb{Z}\)-module consisting of the integers in \(K\) with vanishing trace. The main result of the paper is the formula \[ \zeta(2k+1) = \frac{2q^n-q^{n-1}-q}{q^n-1}\zeta(2k) \lim_{x\to\infty}\frac{j_K(x)}{j_{K,0}(x)}, \] where \(j_K(x)\) is the number of \(A \in I(K)\) with Norm\((A)\leq x\) and \(j_{K,0}(x)\) is the number of \(A \in I(K)\) such that the group \(\mathcal O_{K,0}/(\mathcal O_{K,0} \cap A)\) is of order \(\leq x\). The function \(j_{K,0}(x)\) was introduced and studied by the author in an earlier work [Proc. Intern. Conf. Pure Appl. Math., Beijing, 1994, 141-156; Chin. Sci. Bull. 41, No. 8, 701-702 (1996; Zbl 0876.11051)]. Motivated by the above formula the author now goes on to discuss elementary properties of the function \(j_K(x)/j_{K,0}(x)\), assuming that \(K\) is an algebraic number field subject to various conditions.
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 1375091
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: zeta function for number fields
zbMATH Keywords: cyclotomic field
MaRDI profile type: Publication
full work available at URL: jnth.1999.2408
OpenAlex ID: W1967340551
cites work: Q3735800
DOI: 10.1006/JNTH.1999.2408
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description: scientific article; zbMATH DE number 1375091