10. Let $\sigma_{k}(n)$ denote the sum of the $k$ th powers of the divisors of $n$, so that $\sigma_{k}(n)=\sum_{d \mid n} d^{k}$. Note that $\sigma_{1}(n)=\sigma(n)$.
a) Find $\sigma_{3}(4), \sigma_{3}(6)$ and $\sigma_{3}(12)$.
b) Give a formula for $\sigma_{k}(p)$, where $p$ is prime.
c) Give a formula for $\sigma_{k}\left(p^{a}\right)$, where $p$ is prime, and $a$ is a positive integer.
d) Show that the function $\sigma_{k}$ is multiplicative.
e) Using parts (c) and (d), find a formula for $\sigma_{k}(n)$, where $n$ has prime-power factorization $n=p_{1}^{a_{1}} p_{2}^{a_{2}^{2}} \cdots p_{m}^{a^{a}}$.
10. a) $73,252,2044$
b) $1+p^{k}$
c) $\left(p^{k(a+1)}-1\right) /\left(p^{k}-1\right)$
e) $\prod_{j=1}^{m}\left(p_{j}^{k(a,+1)}-1\right) /\left(p_{j}^{k}-1\right)$