The Sum of Divisors Function

For an integer n, the definition of the sums of divisors function is

d | n

where the sum is over all positive divisors of n.

For example:

= 1 + 2 + 3 + 6 = 12

= 1 + 2 + 4 + 8 = 15

To find a formula for let us

a) determine for p prime

b) Show that the sums of divisors function is multiplicative

a) For a prime power the divisors are

, , , ...,

so that

= 1 + + + + ... +

Note that this is a geometric series by p:

Multiplying the series by p

= + + ... +

and subtracting the original series

- =

solving for

=

>

b) Now we will show that when (m,n) = 1 the sums of the divisors function is multiplicative. In other words

=

A divisor d of can be written uniquely as where | m and | n while ( , ) = 1. Conversely, if | m and | n , then | as (m, n) = 1.

=

d |

=

|

=

| m , | n

=
| m | n

=

Using (a), (b), and the prime factorization of n gives a formula for

prime factorization of n:

= ...

Since the -function is multiplicative by (b)

= ...

Applying =

= * * ... *

Note two special cases:

1. n =

= =

2. n = prime

= =

Maple has a command which we will use in the next section.

> with(numtheory):

> sigma(6);

> divisors(6);

> tau(6);

divisors lists the divisors of n, and tau is the number of divisors.

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