# Math Help - Proving on positive integers?

1. ## Proving on positive integers?

Prove that for all positive integers a and b, σ(ab) ≤ σ (a) σ (b).

2. ## Re: Proving on positive integers?

Is this the sum of divisors function?

3. ## Re: Proving on positive integers?

The sum of divisors function $\sigma$ is multiplicative for relatively prime integers. We will consider any common prime factor (p) of a and b. Suppose that $p^k$ completely divides $a$ and $p^m$ completely divides $b$.

You want to prove that

$\sigma(p^{k+m}) \le \sigma (p^k) \sigma (p^m)$

However, finding the sum of divisors of a prime power is easy (just a geometric series).

$1 + p + \dots + p^{k+m} \le (1 + p + \dots p^k)(1 + p + \dots p^m)$

And you can go on from there.