13. More on Positive Divisors

The outline of the this notebook is as follows:

Today we will spend time looking at the functions \tau which counts the number of positive the divisors of n and \sigma which is the sum of positive divisors of n.  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 f defined on the set of positive integers. That is f : \mathbb{Z}_{>0}\to\mathbb{C}. An arithmetic function f is said to be multiplicative if it is not identically zero and if

f(a\,b) = f(a)\,f(b) \quad \text{ whenever } \gcd(a,\,b) = 1.

Let a and b be positive integers and f : \mathbb{Z}_{>0}\to\mathbb{C} is an arithmetic function. If \gcd(a,\,b)=1 and f(a\,b) = f(a)\,f(b) then f is said to be an multiplicative.

f is said to be completely multiplicative if f(a\,b) = f(a)\,f(b) for all a,b \in \mathbb{N}.

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

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

  1. Is \tau an arithmetic function?

    • Yes \tau is an arithmetic function by definition. In particular, for all n \in \mathbb{N}, \tau(n) \in \mathbb{N} and since \mathbb{N} \subset \mathbb{C}, it follows that \tau is an arithmetic function.

  2. For any two positive integers a and b with \gcd(a,b) = 1, is \tau(a\,b) = \tau(a)\,\tau(b)?

    • We want to count the number of divisors of the product a\,b given that \gcd(a,b)=1. From \gcd(a,b) = 1, we know that a does not divide b and vice-versa. Therefore, for a given divisor of a say a_{1}, the set divisors of the product a\,b will be a_{1} multiplied by all the divisors of b. That is, the set \{a_{1} k \, | \, k \text{ divides } b\} with cardinality \tau(b). Repeating this for all the divisors of a is exactly the product \tau(a) \, \tau(b). This completes the proof.

How will you define \tau 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 \tau was multiplicative. What went wrong here?

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

  • 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 \tau or \sigma for primes.

  • Formula for \tau or \sigma for a power of a prime.

  • General formula for \tau or \sigma.

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

In mathematics, a perfect square say x is the product of some integer say y with itself. That is, x = y^{2}.

  • 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 n^{2}, \tau(n^2) 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 \tau to derive a formula for \sigma for any positive integer n.

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

  • Check if \sigma is multiplicative, that is,

    \sigma(a\,b) = \sigma(a)\,\sigma(b)\qquad \text{whenever } \gcd(a,\,b)=1.

    If it is multiplicative, then try to prove it.

Write your proof in this cell.

How will you define \sigma 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 n, \sigma(n), that we could explore?

  • Write your questions here.