# §27.3 Multiplicative Properties

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 $\displaystyle d\left(n\right)$ $\displaystyle=\prod_{r=1}^{\nu\left(n\right)}(1+a_{r}),$ 27.3.6 $\displaystyle\sigma_{\alpha}\left(n\right)$ $\displaystyle=\prod_{r=1}^{\nu\left(n\right)}\frac{p^{\alpha(1+a_{r})}_{r}-1}{% p^{\alpha}_{r}-1},$ $\alpha\neq 0$.

Related multiplicative properties are

 27.3.7 $\sigma_{\alpha}\left(m\right)\sigma_{\alpha}\left(n\right)=\sum_{d\mathbin{|}% \left(m,n\right)}d^{\alpha}\sigma_{\alpha}\left(\frac{mn}{d^{2}}\right),$
 27.3.8 $\phi\left(m\right)\phi\left(n\right)=\phi\left(mn\right)\phi\left(\left(m,n% \right)\right)/\left(m,n\right).$

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 $f(n)=\prod_{r=1}^{\nu\left(n\right)}\left(f(p_{r})\right)^{a_{r}}.$ ⓘ