# Divisor Functions

Definition. The sum of divisors function is given by

The number of divisors function is given by

Example. Recall that a number is perfect if it's equal to the sum of its divisors other than itself. It follows that a number n is perfect if .

Example.

I want to show that and are multiplicative. I can do most of the work in the following lemma.

Lemma. The divisor sum of a multiplicative function is multiplicative.

Proof. Suppose f is multiplicative, and let be the divisor sum of f. Suppose . Then

Then

Now , so if and , then . Therefore, multiplicativity of f implies

Now every divisor d of can be written as , where and . Going the other way, if and then . So I may set , where , and replace the double sum with a single sum:

This proves that is multiplicative.

Example. The identity function is multiplicative: for all m, n (so a fortiori for ). Therefore, the divisor sum of is multiplicative. But

Hence, the sum of divisors function is multiplicative.

Example. The constant function is multiplicative: for all m, n (so a fortiori for ). Therefore, the divisor sum of I is multiplicative. But

Hence, the number of divisors function is multiplicative.

I'll use multiplicativity to obtain formulas for and in terms of their prime factorizations (as I did with ). First, I'll get the formulas in the case where n is a power of a prime.

Lemma. Let p be prime. Then:

Proof. The divisors of are 1, p, , ..., . So the sum of the divisors is

And since the divisors of are 1, p, , ..., , there are of them, and

Theorem. Let , where the p's are distinct primes and for all i. Then:

Proof. These results follow from the preceding lemma, the fact that and are multiplicative, and the fact that the prime power factors are pairwise relatively prime.

Here is a graph of for .

Note that if p is prime, . This gives the point , which lies on the line . This is the line that you see bounding the dots below.

Here is a graph of for .

If p is prime, . Thus, repeatedly returns to the horizontal line , which you can see bounding the dots below.

Example. , so

Example. For each n, there are only finitely many numbers k whose divisors sum to n: . For k divides itself, so

This says that k must be less than n. So if I'm looking for numbers whose divisors sum to n, I only need to look at numbers less than n. For example, if I want to find all numbers whose divisors sum to 42, I only need to look at .

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