# 12. Positive Divisors¶

The outline of the this notebook is as follows:

Today we will spend time looking at the functions which
counts the number of positive the divisors of and
which is the sum of positive divisors of . Our
goal is, once again, to use our minds, paper and pencil,
and *SageMath* to combine to not just conjecture, but start to give
logical reasons for our conjectures.

Definition: An arithmetic function is real or complex defined on the set of positive integers. That is . An arithmetic function is said to be multiplicative if it is not identically zero and if

Let and be positive integers and is an arithmetic function. If and then is said to be an multiplicative.

is said to be completely multiplicative if for all .

Knowing the definition of and what it means for an arithmetic function to be multiplicative, can we proof that and are both multiplicative function?

Let us first consider . To prove that is a multiplicative function, we must ask ourselves the following questions.

Is an arithmetic function?

Yes is an arithmetic function by definition. In particular, for all and since , it follows that is an arithmetic function.

For any two positive integers and with , is ?

We want to count the number of divisors of the product given that . From , we know that does not divide and vice-versa. Therefore, for a given divisor of say , a set divisors of the product will be multiplied by all the divisors of . That is, the set with cardinality . Repeating this for all the divisors of is exactly the product . This completes the proof.

How will you define in SageMath?