The divisor power sums ${\sigma }_k$

For an integer $k\ge 0$ and a positive integer $n$ let

${\sigma }_k(n):={\sum }_{d\mid n,1\le d\le n}{d}^k,$

the sum of the $k$-th powers of the positive divisors of $n$ (Divisibility in $\mathbb{Z}$: $d\mid a$ when $a=dq$ for some integer $q$, Integer powers $a^m$, Finite sums and finite products, by recursion, The integers as equivalence classes of pairs of naturals). Thus ${\sigma }_0(n)$ is the number of positive divisors and ${\sigma }_1(n)$ is their sum. The sum is over the set of positive divisors of $n$, a finite set: it is nonempty because $1\mid n$, and it is bounded above by $n$ when $n\ge 1$ (If $d\mid a$ and $a\mathrm{\ne }0$ then $d\mathrm{\ne }0$ and $\mathrm{\mid }d\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }$; hence the set of divisors of a nonzero integer is bounded above by $\mathrm{\mid }a\mathrm{\mid }$), so the displayed sum is a finite sum of integers. Each ${\sigma }_k(n)$ is therefore a positive integer: by induction on $k$, $d^0=1>0$ and $d^{k+1}=d^{k}d>0$ for every positive integer $d$, since positive integers are closed under multiplication (Integer powers $a^m$, The principle of mathematical induction, The integers form a totally ordered ring). The divisor $d=1$ contributes $1^k=1$, and adding the other positive integer summands preserves positivity by compatibility of the integer order with addition (The integers form a totally ordered ring). These functions are used only to express the Fourier coefficients of the Eisenstein series on this page.