§27.3 Multiplicative Properties

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

Except for $\mathop{\nu\/}\nolimits\!\left(n\right)$ , $\mathop{\Lambda\/}\nolimits\!\left(n\right)$ , $p_{n}$ , and $\mathop{\pi\/}\nolimits\!\left(x\right)$ , 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:: $m$ : positive integer, $n$ : positive integer and $f(n)$ : multiplicative function
Permalink:: http://dlmf.nist.gov/27.3.E1
Encodings:: TeX, pMML, png

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 $f(n)=\prod _{{r=1}}^{{\mathop{\nu\/}\nolimits\!\left(n\right)}}f(p^{{a_{r}}}_{r}).$

Symbols:: $\mathop{\nu\/}\nolimits\!\left(n\right)$ : number of distinct primes dividing $n$ , $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function
Referenced by:: §27.20, §27.3
Permalink:: http://dlmf.nist.gov/27.3.E2
Encodings:: TeX, pMML, png

In particular,

27.3.3 $\mathop{\phi\/}\nolimits\!\left(n\right)=n\prod _{{p\divides n}}(1-p^{{-1}}),$

Symbols:: $\mathop{\phi _{{k}}\/}\nolimits\!\left(n\right)$ : sum of powers of integers relatively prime to $n$ , $n$ : 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

27.3.4 $\mathop{J_{{k}}\/}\nolimits\!\left(n\right)=n^{k}\prod _{{p\divides n}}(1-p^{{-k}}),$

Symbols:: $\mathop{J_{{k}}\/}\nolimits\!\left(n\right)$ : 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

27.3.5 $\mathop{d\/}\nolimits\!\left(n\right)=\prod _{{r=1}}^{{\mathop{\nu\/}\nolimits\!\left(n\right)}}(1+a_{r}),$

Symbols:: $\mathop{d_{{k}}\/}\nolimits\!\left(n\right)$ : divisor function, $\mathop{\nu\/}\nolimits\!\left(n\right)$ : number of distinct primes dividing $n$ , $n$ : positive integer and $a_{r}$ : positive exponent
Permalink:: http://dlmf.nist.gov/27.3.E5
Encodings:: TeX, pMML, png

27.3.6 $\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(n\right)=\prod _{{r=1}}^{{\mathop{\nu\/}\nolimits\!\left(n\right)}}\frac{p^{{\alpha(1+a_{r})}}_{r}-1}{p^{\alpha}_{r}-1},$ $\alpha\neq 0$ .

Symbols:: $\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(n\right)$ : sum of powers of divisors of $n$ , $\mathop{\nu\/}\nolimits\!\left(n\right)$ : number of distinct primes dividing $n$ , $n$ : 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

Related multiplicative properties are

27.3.7 $\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(m\right)\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(n\right)=\sum _{{d\divides\left(m,n\right)}}d^{\alpha}\mathop{\sigma _{{\alpha}}\/}\nolimits\left(\frac{mn}{d^{2}}\right),$

Symbols:: $\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(n\right)$ : sum of powers of divisors of $n$ , $d$ : positive integer, $m$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.3.E7
Encodings:: TeX, pMML, png

27.3.8 $\mathop{\phi\/}\nolimits\!\left(m\right)\mathop{\phi\/}\nolimits\!\left(n\right)=\mathop{\phi\/}\nolimits\!\left(mn\right)\mathop{\phi\/}\nolimits\!\left(\left(m,n\right)\right)/\left(m,n\right).$

Symbols:: $\mathop{\phi _{{k}}\/}\nolimits\!\left(n\right)$ : sum of powers of integers relatively prime to $n$ , $m$ : positive integer and $n$ : positive integer
Permalink:: http://dlmf.nist.gov/27.3.E8
Encodings:: TeX, pMML, png

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

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

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

27.3.10 $f(n)=\prod _{{r=1}}^{{\mathop{\nu\/}\nolimits\!\left(n\right)}}\left(f(p_{r})\right)^{{a_{r}}}.$

Symbols:: $\mathop{\nu\/}\nolimits\!\left(n\right)$ : number of distinct primes dividing $n$ , $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers, $a_{r}$ : positive exponent and $f(n)$ : multiplicative function
Referenced by:: §27.20
Permalink:: http://dlmf.nist.gov/27.3.E10
Encodings:: TeX, pMML, png