On the asymptotic behaviour of the ideal counting function in quadratic number fields
type: Item
instance of: scholarly article
title: On the asymptotic behaviour of the ideal counting function in quadratic number fields
zbMATH Open document ID: 0713.11071
DOI: 10.1007/BF01501132
author: D. Kharzeev
published in: Monatshefte für Mathematik
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178451
review text: Let \(K\) be a quadratic number field with discriminant \(D\) and denote by \(F(n)\) the number of ideals with norm equal to \(n\). Then \(F(n)=F_{\chi_D}(n):=\sum_{d\mid n}\chi_D(d)\), where \(\chi_D\) is a real primitive Dirichlet character modulo \(| D|\). More generally, consider arbitrary primitive Dirichlet characters \(\chi\) and \(\Psi\) modulo \(k>1\) and the corresponding functions \(F_{\chi}(n)\) and \(F_{\Psi}(n)\). For \(r\geq 1\) the following asymptotic formula is proved \[ \sum_{n\leq x}F_{\chi}(n)F_{\Psi}(n+r) = M_{\chi,\Psi}(r)x + E_{\chi,\Psi}(x,r). \] Here \(M_{\chi,\Psi}(r)\) is an explicitly determined function of \(r\) which depends on \(\chi\) and \(\Psi\), and for every \(\varepsilon >0\) the error term is bounded by \[ | E_{\chi,\Psi}(x,r)| \ll k^{3/2+\varepsilon}X^{5/6+\varepsilon}\] uniformly for \(r \ll k^{1/2}x^{5/6}\). Moreover, \(E_{\chi,\Psi}(x,r)\) is small on average, i.e. \[ \int^{2X}_{X}| E_{\chi,\Psi}(x,r)|^2\,dx \ll k^{4+\varepsilon}X^{5/2+\varepsilon} \] uniformly for \(r \ll kX^{3/4}\).
Mathematics Subject Classification ID: 11R47
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 4173124
zbMATH Keywords: ideal counting function
zbMATH Keywords: generalized Kloosterman sums
zbMATH Keywords: primitive Dirichlet character
zbMATH Keywords: asymptotic formula
MaRDI profile type: Publication
cites work: An Asymptotic Formula in the Theory of Numbers
cites work: On Kloosterman's sum
cites work: The Fourth Power Moment of the Riemann Zeta Function
cites work: Q4752704
cites work: Q3903076
cites work: Q5584672
cites work: Q5769133
cites work: AN EXAMPLE IN THE THEORY OF THE SPECTRUM OF A FUNCTION
cites work: Q3735790
OpenAlex ID: W1977691262
description: scientific article; zbMATH DE number 4173124