Sigma of an Integer

Theorem

Let $n$ be an integer such that $n \ge 2$, with prime decomposition $n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$.

Let $\sigma \left({n}\right)$ be the sigma function of $n$.

That is, let $\sigma \left({n}\right)$ be the sum of all positive divisors of $n$.

Then:

$\displaystyle \sigma \left({n}\right) = \prod_{1 \le i \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

Proof

We have that the Sigma Function is Multiplicative.

From Basic Properties of Multiplicative Function, we have:

$f \left({n}\right) = f \left({p_1^{k_1}}\right) f \left({p_2^{k_2}}\right) \ldots f \left({p_r^{k_r}}\right)$

From Sigma of Power of Prime, we have:

$\displaystyle \sigma \left({p_i^{k_i}}\right) = \frac {p_i^{k_i + 1} - 1} {p_i - 1}$

Hence the result.

$\blacksquare$