On the order of Hecke-Landau zeta functions near the line \(\sigma =1\)
type: Item
instance of: scholarly article
title: On the order of Hecke-Landau zeta functions near the line \(\sigma =1\)
zbMATH Open document ID: 0613.12009
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 1986-01-01T00:00:00Z
review text: Let K be a fixed algebraic number field of degree n over the rationals with discriminant d. There are problems in the theory of algebraic numbers for which we need some information about the growth of the Hecke- Landau zeta-function \(\zeta_ K(\sigma +it,\chi)\), \(\chi\) mod \({\mathfrak q}\), near the line \(\sigma =1\) as \(t\to \infty\). In the present paper the authors derive for the region \(1-(n+1)^{-1}\leq \sigma \leq 1\), \(t\geq 1.1\) the estimate \[ | \zeta_ K(\sigma +it,\chi)| \leq c_ 1\cdot N{\mathfrak q}^{c_ 2}\cdot t^{c_ 3(1-\sigma)^{3/2}}\cdot (\log t)^{2/3}\quad, \] where \(c_ 1\), \(c_ 2\) and \(c_ 3\) can be expressed in terms of n and d with pure numerical constants. Specially, we have \(c_ 2=1.03\cdot 10^ 5\cdot n^ 5\), which seems to be far from best possible. The proof is based on some estimates connected with the extension of I. M. Vinogradov's method to the theory of Hecke-Landau zeta-functions.
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 3991627
zbMATH Keywords: estimating of trigonometric sums
zbMATH Keywords: Hecke-Landau zeta-function
zbMATH Keywords: Vinogradov's method
author: Krystyna Maria Bartz
author: Wlodzimierz Stas
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3991627