# 13. More on 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 , the 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*?

```
def tau(n):
r"""
Returns the number of divisors of $n$.
"""
return number_of_divisors(n)
```

```
tau(48)==tau(8)*tau(6)
```

```
False
```

Oops! What just happened? I thought we just proved that was multiplicative. What went wrong here?

Can you think of questions that we could explore on the number of divisors function ?

Write your questions here.

## 13.1. General Formulas¶

Here is the general hint to follow in order to get a general formula for these arithmetic functions.

Formula for or for primes.

Formula for or for a power of a prime.

General formula for or .

Observe that these are the steps we followed to obtain a general formula for the number of divisors function .

In mathematics, a perfect square say is the product of some integer say with itself. That is, .

Investigate on the number of divisors of perfect squares using

*SageMath*.Observe your data very closely and try and come up with a conjecture on a pattern you think will always be true.

Proof your conjecture.

You are encouraged to work in groups.

Right now, try working on your own or in groups to try to prove the following conjectures.

For a perfect square is odd.

(For a full proof, you will probably need a reason why only certain numbers can divide other numbers.)

Now use the procedure we followed to derive a formula for to derive a formula for for any positive integer .

Let us recal the definition of . It is defined as follows: Given a positive , is the sum of all the positive divisors of .

Check if is

*multiplicative*, that is,If it is multiplicative, then try to prove it.

Write your proof in this cell.

How will you define in *SageMath*?

```
def sigma(n):
"""
Returns the sum of the divisors of `n`
"""
pass
```

Can you formulate questions about the sum of positive divisors of , , that we could explore?

Write your questions here.