1. Multiplicative Functions

Definition (Multiplicative Function).

An Arithmetic Function is called multiplicative if and

Remark.

If is multiplicative then

Definition (Completely Multiplicative Function).

An Arithmetic Function is called Completely multiplicative if

Examples

𝟙 are Multiplicative

are non-multiplicative

Theorem.

Prove is Multiplicative

Proof.
lets show there is a bijection

let we want to find . .

let

Hence
this prove is surjective
now lets prove is injective

hence is bijection
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Proposition.

If and are multiplicative then so is . conversly if and is multiplicative the is multiplicative

Proof.
Write
let we want to show that

we have proved the first part now the second part
suppose is not multiplicative
then there . .

chose such , so that is the smallest possible

Case I suppose
then
now

then is not multiplicative
So
now

as is not 0 then is not multiplicative

Corollary.

Let and such that then is multiplicative is multiplicative

Proposition.

Let and then exist if

Proof.

Suppose exist then

conversely suppose
suppose then we must have

Lets solve it by induction

Suppose is defined
we want to define now we need
such that

this completes the proof
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Examples

  2. find
    as is multiplicative so is so its sufficient to find for all primes and
Remark.

If then

Proof.

□

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