On the divisor function of sets with even partition functions
type: Item
instance of: scholarly article
title: On the divisor function of sets with even partition functions
zbMATH Open document ID: 1121.11071
author: N. Baccar
author: Fethi Ben Saïd
author: Ahlem Zekraoui
published in: Acta Mathematica Hungarica
publication date: 2007-01-08T00:00:00Z
review text: For a set \(\mathcal A\) of positive integers let \(p(\mathcal A,n)\) denote the number of partitions of \(n\) into parts from \(\mathcal A\). \textit{J.-L. Nicolas, I. Z. Ruzsa} and \textit{A. Sárközy} [J. Number Theory 73, 292--317 (1998; Zbl 0921.11050)] proved that if \(\emptyset \neq \mathcal B\subseteq \{1,2,\dots,N\}\) then there exists a unique \(\mathcal A\) such that \(\mathcal A\cap\{1,\dots,N\}=\mathcal B\) and \(p(\mathcal A,n)\) is even for \(n>N\). In other words, for a nonconstant polynomial \(P\in \text{GF}(2)[z]\) with \(P(0)=1\), \(\exists!\mathcal A=\mathcal A(P)\) with the property \(\sum_{n\geq 0}p(\mathcal A,n)z^n\equiv P(z)\pmod 2\). Denote by \((\sigma(\mathcal A,n)\) the sum of divisors of \(n\) belonging to \(\mathcal A\). To determine the elements of \(\mathcal A\), it is important to consider the periodicity of the sequences \(S(\mathcal A,k)=(\sigma(\mathcal A,2^kn)\mod{2^{k+1}})_{n\geq 1}\) for all \(k\geq 0\) which was proved by \textit{F. Ben Saïd} and \textit{J.-L. Nicolas} [Acta Arith. 106, 183--196 (2003; Zbl 1052.11069)]. The values of the above sequences were obtained in terms of the Legendre symbol for \(\mathcal A(1+z+z^3)\) (cf. \textit{F. Ben Saïd} and \textit{J.-L. Nicolas} [Sémin. Lothar. Comb. 46, 25 p. (2002; Zbl 1042.11008)]), where as the discrete logarithm was used for \(\mathcal A(1+z+z^3+z^4+z^5)\) cf. \textit{J.-L. Nicolas} [Period. Math. Hung. 43, 177--189 (2001; Zbl 0980.11049)]. In the paper under review the values of \(S(\mathcal A,k)\) are given in terms of orbits under the multiplicative action of \(\langle 2\rangle\leq (Z/\beta Z)^*\) on \(Z/\beta Z\) for odd \(\beta \geq 3\) when \(\text{ord}(P)=2^\alpha\beta\) (the minimal positive integer \(m\) for which \(P(z)\) divides \(1+z^m\) in \(\text{GF}(2)[z]\)).
reviewed by: Mihály Szalay
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11B50
zbMATH DE Number: 5082275
zbMATH Keywords: partitions
zbMATH Keywords: periodic sequences
zbMATH Keywords: order of polynomial
zbMATH Keywords: orbits
zbMATH Keywords: symmetric functions
zbMATH Keywords: cyclotomic polynomials
zbMATH Keywords: Graeffe transformation
MaRDI profile type: Publication
full work available at URL: s10474-006-0062-4
OpenAlex ID: W2032156832
DOI: 10.1007/S10474-006-0062-4
description: scientific article; zbMATH DE number 5082275