Some restricted partition functions
type: Item
instance of: scholarly article
title: Some restricted partition functions
zbMATH Open document ID: 0788.11043
author: Peter Borwein
published in: Journal of Number Theory
publication date: 1994-06-05T00:00:00Z
review text: Let \(\| f(q)\|_ A= \sup_{q\in A} | f(q)|\). The problem discussed in this paper is the size of \[ \eta(n,p)= \min_{\textstyle {{{\alpha_ 1, \alpha_ 2,\dots, \alpha_ n\in\mathbb{N}} \atop {p\nmid \alpha_ i}}}} \Biggl\| \prod_{k=1}^ n (1-q^{\alpha_ k}) \Biggr\|_{| q|=1}, \] where \(p\) is prime. Previously, Erdős and Szekeres had raised the problem of estimating the above expression but without the condition \(p\nmid \alpha_ k\). The main result proved here is that \(\eta(n,p)\geq p^{n/(p-1)}\) for all primes \(p\), and that \(\eta(n,p)= O(p^{n/(p-1)})\) holds only for \(p\leq 13\). This distinction between primes \(\leq 13\) and those \(>13\) arises by a study of the product \(\displaystyle{\prod^ n_{{{k=1} \atop {p\nmid k}}}}(1-q^ k)\) which is related to partitions into distinct parts not divisible by \(p\). In the course of his study, the author makes clever use of various trigonometrical sums and products one of which is \(\prod_{k=1}^ n \sin(k\theta+\gamma)\).
reviewed by: Krishnaswami Alladi
Mathematics Subject Classification ID: 11P82
Mathematics Subject Classification ID: 30A10
zbMATH DE Number: 484045
zbMATH Keywords: supremum norm
zbMATH Keywords: asymptotic estimation
zbMATH Keywords: partitions into distinct parts
MaRDI profile type: Publication
full work available at URL: jnth.1993.1074
OpenAlex ID: W2003427440
DOI: 10.1006/JNTH.1993.1074
description: scientific article; zbMATH DE number 484045