# criterion for a multiplicative function to be completely multiplicative

###### Theorem.

Let $f$ be a multiplicative function^{} with convolution inverse $g$. Then $f$ is completely multiplicative if and only if $g\mathit{}\mathrm{(}{p}^{k}\mathrm{)}\mathrm{=}\mathrm{0}$ for all primes $p$ and for all $k\mathrm{\in}\mathrm{N}$ with $k\mathrm{>}\mathrm{1}$.

###### Proof.

Note first that, since $f(1)=1$ and $f*g=\epsilon $, where $\epsilon $ denotes the convolution identity function, then $g(1)=1$. Let $p$ be any prime. Then

$$0=\epsilon (p)=(f*g)(p)=f(1)g(p)+f(p)g(1)=g(p)+f(p).$$ |

Thus, $g(p)=-f(p)$.

Assume that $f$ is completely multiplicative. The statement about $g$ will be proven by induction^{} on $k$. Note that:

$\begin{array}{cc}0\hfill & =\epsilon ({p}^{2})\hfill \\ & =(f*g)({p}^{2})\hfill \\ & =f(1)g({p}^{2})+f(p)g(p)+f({p}^{2})g(1)\hfill \\ & =g({p}^{2})+f(p)(-f(p))+{(f(p))}^{2}\hfill \\ & =g({p}^{2})\hfill \end{array}$

Let $m\in \mathbb{N}$ with $m>2$ such that, for all $k\in \mathbb{N}$ with $$, $g({p}^{k})=0$. Then:

$\begin{array}{cc}0\hfill & =\epsilon ({p}^{m})\hfill \\ & =(f*g)({p}^{m})\hfill \\ & =f(1)g({p}^{m})+f({p}^{m-1})g(p)+f({p}^{m})g(1)\hfill \\ & =g({p}^{m})+{(f(p))}^{m-1}(-f(p))+{(f(p))}^{m}\hfill \\ & =g({p}^{m})\hfill \end{array}$

Conversely, assume that $g({p}^{k})=0$ for all $k\in \mathbb{N}$ with $k>1$. The statement $f({p}^{k})={(f(p))}^{k}$ will be proven by induction on $k$. The statement is obvious for $k=1$. Let $m\in \mathbb{N}$ such that $f({p}^{m-1})={(f(p))}^{m-1}$. Then:

$\begin{array}{cc}0\hfill & =\epsilon ({p}^{m})\hfill \\ & =(f*g)({p}^{m})\hfill \\ & =f({p}^{m-1})g(p)+f({p}^{m})g(1)\hfill \\ & ={(f(p))}^{m-1}(-f(p))+f({p}^{m})\hfill \\ & =-{(f(p))}^{m}+f({p}^{m})\hfill \end{array}$

Thus, $f({p}^{m})={(f(p))}^{m}$. It follows that $f$ is completely multiplicative. ∎

Title | criterion for a multiplicative function to be completely multiplicative |
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Canonical name | CriterionForAMultiplicativeFunctionToBeCompletelyMultiplicative |

Date of creation | 2013-03-22 15:58:44 |

Last modified on | 2013-03-22 15:58:44 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 9 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 11A25 |

Related topic | FormulaForTheConvolutionInverseOfACompletelyMultiplicativeFunction |