:tocdepth: 2

.. _more_on_positive_divisors:

More on Positive Divisors
============================================================================

.. contents:: :local:

.. highlight:: python

The outline of the this notebook is as follows:

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

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

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

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

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

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

.. raw:: html

   <ol>

.. raw:: html

   <li>

Is :math:`\tau` an arithmetic function?

.. raw:: html

   </li>

.. raw:: html

   <ul>

.. raw:: html

   <li>

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

.. raw:: html

   </li>

.. raw:: html

   </ul>

.. raw:: html

   <li>

For any two positive integers :math:`a` and :math:`b` with
:math:`\gcd(a,b) = 1`, is :math:`\tau(a\,b) = \tau(a)\,\tau(b)`?

.. raw:: html

   <ul>

.. raw:: html

   <li>

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

.. raw:: html

   </li>

.. raw:: html

   </ul>

.. raw:: html

   </li>

.. raw:: html

   </ol>

How will you define :math:`\tau` in *SageMath*?

.. code:: ipython3

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


.. code:: ipython3

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




.. parsed-literal::

    False



Oops! What just happened? I thought we just proved that :math:`\tau` was
multiplicative. What went wrong here?



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

-  Write your questions here.

General Formulas
~~~~~~~~~~~~~~~~

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

-  Formula for :math:`\tau` or :math:`\sigma` for primes.
-  Formula for :math:`\tau` or :math:`\sigma` for a power of a prime.
-  General formula for :math:`\tau` or :math:`\sigma`.

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

In mathematics, a perfect square say :math:`x` is the product of some
integer say :math:`y` with itself. That is, :math:`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 :math:`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 :math:`\tau`
to derive a formula for :math:`\sigma` for any positive integer
:math:`n`.

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

-  Check if :math:`\sigma` is *multiplicative*, that is,

   .. math:: \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 :math:`\sigma` in *SageMath*?

.. code:: ipython3

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

Can you formulate questions about the sum of positive divisors of
:math:`n`, :math:`\sigma(n)`, that we could explore?

-  Write your questions here.