Summatory formula of the convolution of two arithmetical functions
type: Item
instance of: scholarly article
title: Summatory formula of the convolution of two arithmetical functions
zbMATH Open document ID: 1012.11089
author: Yuk-Kam Lau
published in: Monatshefte für Mathematik
publication date: 2002-09-10T00:00:00Z
review text: The author proves a very general theorem (too complicated to be fully reproduced here) which furnishes the asymptotic formula for the summatory function of the convolution function \[ f\star g(n) = \sum_{d\mid n}f(d)g\left({n\over d}\right), \] provided sufficient information is available for the summatory functions of \(f(n), g(n)\) and their corresponding generating Dirichlet series. The proof starts by an application of the classical hyperbola method, and then proceeds by methods of complex integration. As an application, the author obtains two asymptotic formulas involving \(d(n) \equiv d_2(n)\), the number of divisors function. The first gives a sharp asymptotic formula for the summatory function of \(d(n)z^{\Omega(n)-\omega(n)}\), which is used to deduce the asymptotic formula for the summatory function of \(d(n)\) subject to the condition \( \Omega(n)-\omega(n) = q\), where \(\Omega(n)\) denotes the number of all prime factors of \(n\), \(\omega(n)\) denotes the number of distinct prime factors of \(n\), and \(z\) is a complex number such that \(|z|\leq R\) with \(R < 4/3\). The author remarks that the latter sum is a special case of the more general sum \[ \sum_{n\leq x,\;\Omega(n)-\omega(n) = q}d_k(n),\quad d_k(n) = \sum_{n_1n_2\cdots n_k=n}1, \] investigated by the reviewer [Publ. Inst. Math. 41, 31-41 (1987; Zbl 0648.10028)].
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 1798918
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: summatory functions
zbMATH Keywords: asymptotic formula
MaRDI profile type: Publication
full work available at URL: s006050200032
OpenAlex ID: W2092522283
DOI: 10.1007/S006050200032
Recommended article: Q5313938
Recommended article: Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions
Recommended article: Q3765863
Recommended article: CONVOLUTION FORMULA AS A STIELTJES RESULTANT
Recommended article: Q3490092
Recommended article: Q5385685
Recommended article: The average behaviour of the error term in a new kind of the divisor problem
Recommended article: The binary additive divisor problem
Recommended article: An Omega Theorem for a Class of Arithmetic Functions
Recommended article: Q3497127
description: scientific article; zbMATH DE number 1798918