On an arithmetical function
type: Item
instance of: scholarly article
title: On an arithmetical function
zbMATH Open document ID: 1071.11005
author: Hershel M. Farkas
published in: The Ramanujan Journal
publication date: 2005-04-07T00:00:00Z
review text: For the positive integer \(n\) one denotes by \(d(n)\) the number of its positive divisors, and by \(\sigma(n)\) their sum. \(\delta(n)\) denotes the difference between the number of those positive divisors of \(n\) which are congruent to \(1\pmod 3\) and the number of those positive divisors of \(n\) which are congruent to \(-1\pmod 3\); \(\delta\) is a multiplicative function and \(\delta(3n+2)=0\). The function \(\delta\) is interesting because of its tight connection to the functions \(\sigma\) and \(d\). Most of the results obtained in the paper are non-elementary. Some of the proofs use the function \[ f(z,y)= \exp \biggl(\frac{\pi i}{6} \biggr)y^{\frac{1}{24}}z^{\frac{1}{6}} \prod_{n=0}^{\infty}(1-y^{3n+3})(1-y^{3n+2}z)(1-y^{3n+1}/z), \] a product representation of the first order theta function [see \textit{H. M. Farkas} and \textit{I. Kra}, Theta Constants, Riemann Surfaces and the Modular Group. Graduate Studies in Mathematics. 37. Providence, RI: (American Mathematical Society) (AMS) (2001; Zbl 0982.30001)]. For \(n=3^\alpha\cdot m, \alpha\geq0\), the author denotes by \(\sigma'(n)\) the number \(\sigma(m)\) and proves the relation \(\delta(n)=3\sum_{j=1}^{n-1}\delta(j)\delta(n-j)=\sigma'(n)\). (Theorem 1). For \(a_n=\sum_{d| n}\frac{n}{d}\) with \((d,3)=1\), the paper contains the proof of several identities, among which we emphasize evaluations for \(\sum_{n=1}^{\infty}a_ny^n\) and \(3a_n-2\sigma(n)=\delta(n)+3\sum_{j=1}^{n-1}\delta(j)\delta(n-j)\), with interesting consequences such as: `\(3n+1\) is prime iff \(\sum_{j=0}^{n-1}\delta(3j+1)\delta(n-j)=n/2\). \(3n+2\) is prime iff \(\sum_{j=0}^n\delta(3j+1)\delta(3(n-j)+1)=n+1\).' The paper ends with a remark concerning the use of the function \(\delta\) to evaluate the number of integer solutions of the equation \(x^2+3y^2=N^2\).
reviewed by: Laurenţiu Panaitopol
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 33B99
zbMATH DE Number: 2153719
zbMATH Keywords: theta functions
zbMATH Keywords: divisor functions
MaRDI profile type: Publication
full work available at URL: s11139-004-0141-5
OpenAlex ID: W1631668759
DOI: 10.1007/S11139-004-0141-5
description: scientific article; zbMATH DE number 2153719