Divisor function

In mathematics the divisor function σ_a(n) is defined as the sum of the ith powers of the divisors of n, or

$\sigma _{a}(n)=\sum _{d|n}d^{a}.$

The divisor function is multiplicative, but not completely multiplicative.

Two Dirichlet series involving the divisor function are:

$\sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\zeta (s-a)$

and

${\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}}$