On exponential sums involving the ideal counting function in quadratic number fields
type: Item
instance of: scholarly article
title: On exponential sums involving the ideal counting function in quadratic number fields
zbMATH Open document ID: 1014.11057
author: Jun Furuya
published in: Monatshefte für Mathematik
publication date: 2002-12-01T00:00:00Z
review text: Let \(K\) be a quadratic number field with discriminant \(D\) and \(\chi\) an arbitrary primitive Dirichlet character modulo \(k>1\). Further let \(F_\chi(n)= \sum_{d|n}\chi(d)\). The author considers exponential sums involving the arithmetical function \(F_\chi(n)\), namely \[ R\bigl(x; \tfrac hq\bigr)= \sum_{n\leq x} F_\chi(n) e^{2\pi ihn/q}, \] where \(h\), \(q\) are co-prime integers with \(q\geq 1\). The remainder term in an asymptotic expansion of this function can be described as \[ P \bigl(x; \tfrac hq\bigr)= R\bigl(x; \tfrac hq\bigr)- x \;\underset {s=1} {\text{Res}} F_\chi \bigl(s; \tfrac hq\bigr)- F_\chi\bigl(0; \tfrac hq\bigr). \] Here \(F_\chi (s; \frac hq)\) means the corresponding generating function \[ F_\chi \bigl(s; \tfrac hq \bigr)= \sum_{n=1}^\infty F_\chi(n) e^{2\pi ihn/q} n^{-s}. \] Now the asymptotic behaviour of \(P(x; \frac hp)\) is studied. A nontrivial upper bound and mean square formulas are given. Furthermore, a sum formula of Voronoi type is developed, that is a representation of \[ \sum_{a< n\leq b} F_\chi(n) e^{2\pi ihn/q} f(n) \] by a convergent infinite series. This generalizes the main result of Chapter 1 in the book of \textit{M. Jutila} [Lecture on a method in the theory of exponential sums, Tata Institute of Fundamental Research, Springer (1987; Zbl 0671.10031)].
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11R47
Mathematics Subject Classification ID: 11L03
zbMATH DE Number: 1837391
zbMATH Keywords: exponential sum
zbMATH Keywords: ideal counting function
zbMATH Keywords: Voronoi summation formula
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: upper bound
zbMATH Keywords: mean square
reviewed by: Ekkehard Krätzel
MaRDI profile type: Publication
full work available at URL: s00605-002-0495-y
OpenAlex ID: W1993994330
DOI: 10.1007/S00605-002-0495-Y
Recommended article: On the mean square of the error terms corresponding to exponential sums involving the ideal counting function
Recommended article: Q4855911
Recommended article: Exponential sums of the form Σχ(x)axζmbx
Recommended article: Q5287742
Recommended article: Q4271933
Recommended article: ON DIRICHLET CHARACTERS OF POLYNOMIAL
Recommended article: Lectures on a method in the theory of exponential sums
Recommended article: The Asymptotic Distribution of Exponential Sums, I
Recommended article: Q3255469
Recommended article: On the fourth power mean of a sum analogues to character sums over short intervals
description: scientific article; zbMATH DE number 1837391