On the error term concerning the number of subgroups of the groups \(\mathbb{Z}_m \times \mathbb{Z}_n\) with \(mn \leq x\)
type: Item
instance of: scholarly article
title: On the error term concerning the number of subgroups of the groups \(\mathbb{Z}_m \times \mathbb{Z}_n\) with \(mn \leq x\)
zbMATH Open document ID: 1455.11130
author: Yankun Sui
author: Dan Liu
published in: Journal of Number Theory
publication date: 2020-08-03T00:00:00Z
review text: For positive integers \(m,n\), let \(s(m,n)\) and \(c(m,n)\) denote respectively, the number of subgroups and the number of cyclic subgroups of \(\mathbb{Z}_m\times \mathbb{Z}_n\). For \(x\geq 1\), define \(D_s(x):=\sum_{mn \leq x}s(m,n)\) and \(D_c(x):=\sum_{mn \leq x}c(m,n)\). The authors prove the following asymptotic formulae by using the method of exponential sums: \[ D_s(x)=xP_4(\log x)+ O(x^{\frac 23}\log^6x),\quad D_c(x)=xR_4(\log x)+ O(x^{\frac 23}\log^6x) \] where \(P_4(u)\) and \(R_4(u)\) are polynomials in \(u\) of degree 4 with leading coefficients \(\frac 1{8\pi^2}\) and \(\frac 3{4\pi^4}\) respectively. Now consider the error terms of the asymptotic formulae of \(D_s(x),D_c(x)\) given by \(\Delta_s(x)=D_s(x)-xP_4(\log x)\), \(\Delta_c(x)=D_c(x)-xR_4(\log x)\). The authors prove the following mean square estimates: for any real \(T\geq 10\), \[ \int_{1}^{T} \Delta_s^2(x)\,dx \ll T^{\frac {77}{36}+\varepsilon},\quad \int_{1}^{T} \Delta_c^2(x)\,dx \ll T^{\frac {77}{36}+\varepsilon}. \]
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 7226991
zbMATH Keywords: number of subgroups
zbMATH Keywords: number of cyclic subgroups
zbMATH Keywords: Dirichlet series
zbMATH Keywords: Perron formula
zbMATH Keywords: asymptotic formula
zbMATH Keywords: error term
zbMATH Keywords: exponential sum
zbMATH Keywords: mean square
reviewed by: C. P. Anil Kumar
MaRDI profile type: Publication
full work available at URL: j.jnt.2020.03.001
OpenAlex ID: W3021094081
cites work: Zeta function of subgroups of abelian groups and average orders
cites work: A zero-density theorem for the Riemann zeta-function
cites work: Q4023619
cites work: ON THE AVERAGE NUMBER OF SUBGROUPS OF THE GROUP ℤm × ℤn
cites work: On estimations of trigonometric sums over primes in short intervals. III
cites work: Q3240889
cites work: On the error term concerning the number of subgroups of the groups $\mathbb {Z}_m \times \mathbb {Z}_n$ with $m,n\le x$
cites work: Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function
cites work: Many-Dimensional Generalized Divisor Problems
DOI: 10.1016/J.JNT.2020.03.001
description: scientific article; zbMATH DE number 7226991