scholarly journals Properties of rational arithmetic functions

2005 ◽  
Vol 2005 (24) ◽  
pp. 3997-4017 ◽  
Author(s):  
Vichian Laohakosol ◽  
Nittiya Pabhapote
Keyword(s):  
Arithmetic Functions ◽  
Multiplicative Functions ◽  
Binomial Type ◽  
Dirichlet Convolution ◽  
Rational Arithmetic

Rational arithmetic functions are arithmetic functions of the formg1∗⋯∗gr∗h1−1∗⋯∗hs−1, wheregi,hjare completely multiplicative functions and∗denotes the Dirichlet convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type identities are investigated; third, binomial-type identities are derived; and finally, properties of the Kesava Menon norm of such functions are proved.

scholarly journals On a subgroup of the group of multiplicative arithmetic functions

1975 ◽  
Vol 20 (3) ◽  
pp. 348-358 ◽  
Author(s):  
T. B. Carroll ◽  
A. A. Gioia
Keyword(s):  
Abelian Group ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Dirichlet Convolution ◽  
Image Position ◽  
Positive Integers ◽  
Multiplicative Arithmetic

An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

scholarly journals A Proof of an Identity for Multiplicative Functions

1979 ◽  
Vol 22 (3) ◽  
pp. 299-304
Author(s):  
K. Krishna
Keyword(s):  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Multiplicative Functions ◽  
Prime Factors ◽  
Dirichlet Convolution

An arithmetic function f is said to be multiplicative if f(mn) = f(m)f(n), whenever (m, n) = 1 and f(1) = 1. The Dirichlet convolution of two arithmetic functions f and g, denoted by f • g, is defined by f • g(n) = Σd|nf(d)g(n/d). Let w(n) denote the product of the distinct prime factors of n, with w(l) = 1. R.

scholarly journals Factorization and arithmetic functions for orders in composition algebras

1973 ◽  
Vol 14 (1) ◽  
pp. 86-95 ◽  
Author(s):  
P. J. C. Lamont
Keyword(s):  
Division Algebra ◽  
Inversion Formula ◽  
Arithmetic Functions ◽  
Convolution Product ◽  
Multiplicative Functions ◽  
Composition Algebras ◽  
Dirichlet Convolution

A well-known product, referred to as the Dirichlet convolution product, is generalized to arithmetic functions defined on an order in a Cayley division algebra. Factorization results for orders, multiplicative functions and analogues of the Moebius inversion formula are discussed.

Additive and multiplicative functions from Nottingham to Urbana-Champaign

2015 ◽  
Vol 11 (05) ◽  
pp. 1357-1366
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Arithmetic Functions ◽  
Multiplicative Functions ◽  
Joint Work ◽  
Multiplicative Arithmetic

The author remembers Heini Halberstam and views their early joint work through the lens of additive and multiplicative arithmetic functions.

scholarly journals Some characterizations of totients

1996 ◽  
Vol 19 (2) ◽  
pp. 209-217 ◽  
Author(s):  
Pentti Haukkanen
Keyword(s):  
Multiplicative Function ◽  
Arithmetical Function ◽  
Multiplicative Functions ◽  
Dirichlet Convolution

An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Euler's phi-function is a famous example of a totient. All completely multiplicative functions are also totients. There is a large number of characterizations of completely multiplicative functions in the literature, while characterizations of totients have not been widely studied in the literature. In this paper we present several arithmetical identities serving as characterizations of totients. We also introduce a new concrete example of a totient.

MORE ON A CERTAIN ARITHMETICAL DETERMINANT

2017 ◽  
Vol 97 (1) ◽  
pp. 15-25 ◽  
Author(s):  
ZONGBING LIN ◽  
SIAO HONG
Keyword(s):  
Linear Algebra ◽  
Acta Math ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Mobius Function ◽  
Dirichlet Convolution ◽  
Positive Integers

Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$, where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$. In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

scholarly journals Completely multiplicative functions arising from simple operations

2004 ◽  
Vol 2004 (9) ◽  
pp. 431-441
Author(s):  
Vichian Laohakosol ◽  
Nittiya Pabhapote
Keyword(s):  
Arithmetic Functions ◽  
Multiplicative Functions ◽  
Convolution Powers ◽  
Multiplicative Arithmetic

Given two multiplicative arithmetic functions, various conditions for their convolution, powers, and logarithms to be completely multiplicative, based on values at the primes, are derived together with their applications.

scholarly journals On the periodicity of a class of arithmetic functions associated with multiplicative functions

2013 ◽  
Vol 133 (6) ◽  
pp. 2005-2020 ◽  
Author(s):  
Guoyou Qian ◽  
Qianrong Tan ◽  
Shaofang Hong
Keyword(s):  
Arithmetic Functions ◽  
Multiplicative Functions

scholarly journals On the usual product of rational arithmetic functions

1990 ◽  
Vol 59 (2) ◽  
pp. 191-196 ◽  
Author(s):  
Pentti Haukkanen ◽  
Jerzy Rutkowski
Keyword(s):  
Arithmetic Functions ◽  
Rational Arithmetic ◽  
Usual Product

scholarly journals TWO GENERALIZATIONS OF THE BUSCHE–RAMANUJAN IDENTITIES

2013 ◽  
Vol 09 (05) ◽  
pp. 1301-1311 ◽  
Author(s):  
LÁSZLÓ TÓTH
Keyword(s):  
Dirichlet Series ◽  
Arithmetic Functions ◽  
Symmetric Polynomials ◽  
Several Variables ◽  
Dirichlet Convolution ◽  
Multiple Dirichlet Series

We derive two new generalizations of the Busche–Ramanujan identities involving the multiple Dirichlet convolution of arithmetic functions of several variables. The proofs use formal multiple Dirichlet series and properties of symmetric polynomials of several variables.

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