 Generalized divisor function

ID: generalized-divisor-function

k \geq 1

, the generalized divisor function

d_{k} (n)

n

into

k

n = \prod_{p} p^{a_{p}}

, then

d_{k} (n) = \prod_{p ∣ n} (k - 1 a _{p} + k - 1) .

To 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) = (2 4) (2 3) = 18

. The ordered condition matters: exchanging two unequal factors changes the counted tuple. Absolute convergence for

Re s > 1

ζ (s)^{k} = \sum_{n \geq 1} d_{k} (n) n^{- s}

and connecting these counts with powers of the Riemann zeta function.

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