Generalized divisor function ($d_k(n)=\#\{(a_1,\dots ,a_k)\in {\mathbb{Z}}_{>0}^k:a_1\cdots a_k=n\}$)

For an integer $k\ge 1$, the generalized divisor function $d_k(n)$ counts ordered factorizations of a positive integer $n$ into $k$ positive integer factors. If $n={\prod }_p{p}^{a_p}$, thenTo prove the formula, distribute the exponent $a_p$ among the $k$ factors. The number of nonnegative exponent tuples with that sum is the stars and bars count, and different primes can be assigned independently.

Thus $d_1(n)=1$ and $d_2(n)$ is the ordinary divisor function. For $12=2^2\cdot 3$, $d_3(12)=\left(\genfrac{}{}{0ex}{}{4}{2}\right)\left(\genfrac{}{}{0ex}{}{3}{2}\right)=18$. The ordered condition matters: exchanging two unequal factors changes the counted tuple. Absolute convergence for $\mathrm{Re}s>1$ permits multiplying Dirichlet series, giving $\zeta (s{)}^k={\sum }_{n\ge 1}{d}_k(n)n^{-s}$ and connecting these counts with powers of the Riemann zeta function.