The Euler-Kronecker constant of a cyclotomic field
type: Item
instance of: scholarly article
zbMATH Open document ID: 1259.11105
author: V. Kumar Murty
publication date: 2012-05-30T00:00:00Z
full work available at URL: 35-2.html
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 6039661
zbMATH Keywords: Euler-Kronecker constant
zbMATH Keywords: cyclotomic field
zbMATH Keywords: Ihara's conjecture
MaRDI profile type: Publication
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title: The Euler-Kronecker constant of a cyclotomic field
published in: Annales des Sciences Mathématiques du Québec
review text: Let \(K\) be a number field and let \(\zeta_K(s)\) be its Dedekind \(\zeta\)-function, which has a simple pole at \(s = 1\). The Laurent expansion at \(s = 1\) therefore has the form NEWLINE\[NEWLINE\zeta_K(s) = c_{-1}(s - 1)^{-1} + c_0 + c_1(s - 1) + \cdotsNEWLINE\]NEWLINE with \(c_{-1} \not= 0\). The Euler-Kronecker constant of \(K\) is the real number \(\gamma_K = c_0/c_{-1}\). The article under review explores the behaviour of \(\gamma_K\) as \(K\) varies over cyclotomic fields of prime conductor. See below for a more precise description of the main result.NEWLINENEWLINESome bounds on \(\gamma_K\) have been proven under the Generalized Riemann Hypothesis (GRH). For example, Ihara has shown that the GRH implies the existence of positive constants \(c_1\) and \(c_2\) such that, for any number field \(K \not= \mathbb{Q}\), NEWLINE\[NEWLINE -c_1 \log d_K \leq \gamma_K \leq c_2 \log \log d_K ,NEWLINE\]NEWLINE where \(d_K\) is the absolute value of the discriminant of \(K\).NEWLINENEWLINEWe consider a conjecture of Ihara on Euler-Kronecker constants for cyclotomic fields. For a positive integer \(m\), let \(\gamma_m\) be the Euler-Kronecker constant of the field \(K_m = \mathbb{Q}(\zeta_m)\) where \(\zeta_m\) is a primitive \(m\)th root of unity. Then Ihara conjectures that there are constants \(c_3\) and \(c_4\) lying in the interval \((0,2]\) such that, for any \(\varepsilon > 0\), there is a positive integer \(N\) with the property that NEWLINE\[NEWLINE (c_3 - \varepsilon) \log m < \gamma_m < (c_4 + \varepsilon) \log m NEWLINE\]NEWLINE holds for \(m \geq N\).NEWLINENEWLINEThe main result of the present article may be stated as follows: NEWLINE\[NEWLINE \sum_{\frac{1}{2} Q < q \leq Q} |\gamma_q| \ll \pi^*(Q) \log Q ,NEWLINE\]NEWLINE where the sum runs over primes in the interval \((\frac{1}{2}Q,Q]\) and \(\pi^*(Q)\) denotes the number of primes in that interval. Thus, the article demonstrates Ihara's upper bound on average (in the prime-conductor situation) and does…
reviewed by: Paul Buckingham
description: scientific article; zbMATH DE number 6039661