27 Functions of Number Theory Multiplicative Number Theory 27.2 Functions 27.4 Euler Products and Dirichlet Series

§27.3 Multiplicative Properties

ⓘ

Keywords:: completely multiplicative, completely multiplicative functions, multiplicative, multiplicative functions, multiplicative number theory, number-theoretic functions
Notes:: See Apostol (1976, pp. 27–50).
Permalink:: http://dlmf.nist.gov/27.3
See also:: Annotations for Ch.27

Except for $\nu\left(n\right)$ , $\Lambda\left(n\right)$ , $p_{n}$ , and $\pi\left(x\right)$ , the functions in §27.2 are multiplicative, which means $f(1)=1$ and

27.3.1		$f(mn)=f(m)f(n),$
$\left(m,n\right)=1$ .
ⓘ Symbols: $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $m$ : positive integer, $n$ : positive integer and $f(n)$ : multiplicative function Permalink: http://dlmf.nist.gov/27.3.E1 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27

If $f$ is multiplicative, then the values $f(n)$ for $n>1$ are determined by the values at the prime powers. Specifically, if $n$ is factored as in (27.2.1), then

27.3.2		$f(n)=\prod_{r=1}^{\nu\left(n\right)}f(p^{a_{r}}_{r}).$
ⓘ Symbols: $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function Referenced by: §27.20, §27.20, §27.3 Permalink: http://dlmf.nist.gov/27.3.E2 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27

In particular,

27.3.3	$\displaystyle\phi\left(n\right)$	$\displaystyle=n\prod_{p\mathbin{\|}n}(1-p^{-1}),$
ⓘ Symbols: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers A&S Ref: 24.3.2 I.C Permalink: http://dlmf.nist.gov/27.3.E3 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27
27.3.4	$\displaystyle J_{k}\left(n\right)$	$\displaystyle=n^{k}\prod_{p\mathbin{\|}n}(1-p^{-k}),$
ⓘ Symbols: $J_{\NVar{k}}\left(\NVar{n}\right)$ : Jordan’s function, $k$ : positive integer, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers Permalink: http://dlmf.nist.gov/27.3.E4 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27
27.3.5	$\displaystyle d\left(n\right)$	$\displaystyle=\prod_{r=1}^{\nu\left(n\right)}(1+a_{r}),$
ⓘ Symbols: $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer and $a_{r}$ : positive exponent Permalink: http://dlmf.nist.gov/27.3.E5 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27
27.3.6	$\displaystyle\sigma_{\alpha}\left(n\right)$	$\displaystyle=\prod_{r=1}^{\nu\left(n\right)}\frac{p^{\alpha(1+a_{r})}_{r}-1}{% p^{\alpha}_{r}-1},$
$\alpha\neq 0$ .
ⓘ Symbols: $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$ : sum of powers of divisors of a number, $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers and $a_{r}$ : positive exponent A&S Ref: 24.3.3 I.C Permalink: http://dlmf.nist.gov/27.3.E6 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27

Related multiplicative properties are

27.3.7		$\sigma_{\alpha}\left(m\right)\sigma_{\alpha}\left(n\right)=\sum_{d\mathbin{\|}% \left(m,n\right)}d^{\alpha}\sigma_{\alpha}\left(\frac{mn}{d^{2}}\right),$
ⓘ Symbols: $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$ : sum of powers of divisors of a number, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $d$ : positive integer, $m$ : positive integer and $n$ : positive integer Permalink: http://dlmf.nist.gov/27.3.E7 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27

27.3.8		$\phi\left(m\right)\phi\left(n\right)=\phi\left(mn\right)\phi\left(\left(m,n% \right)\right)/\left(m,n\right).$
ⓘ Symbols: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $m$ : positive integer and $n$ : positive integer Permalink: http://dlmf.nist.gov/27.3.E8 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27

A function $f$ is completely multiplicative if $f(1)=1$ and

27.3.9		$f(mn)=f(m)f(n),$
$m,n=1,2,\dots$ .
ⓘ Symbols: $m$ : positive integer, $n$ : positive integer and $f(n)$ : multiplicative function Permalink: http://dlmf.nist.gov/27.3.E9 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27

Examples are $\left\lfloor 1/n\right\rfloor$ and $\lambda\left(n\right)$ , and the Dirichlet characters, defined in §27.8.

If $f$ is completely multiplicative, then (27.3.2) becomes

27.3.10		$f(n)=\prod_{r=1}^{\nu\left(n\right)}\left(f(p_{r})\right)^{a_{r}}.$
ⓘ Symbols: $\nu\left(\NVar{n}\right)$ : number of distinct primes dividing a number, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function Referenced by: §27.20, §27.20 Permalink: http://dlmf.nist.gov/27.3.E10 Encodings: TeX, pMML, png See also: Annotations for §27.3 and Ch.27

27.2 Functions 27.4 Euler Products and Dirichlet Series

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