Square of Riemann Zeta Function

Theorem

 * $\ds \map {\zeta^2} z = \sum_{k \mathop = 1}^\infty \frac {\map {\sigma_0} k} {k^z}$

where:
 * $\zeta$ is the Riemann zeta function
 * $\sigma_0$ is the divisor counting function.

Proof
Expanding this product, we get:

We see that each $\dfrac 1 {n^z}$ term in this sum will occur as many times as there are ways represent $n$ as $a b$, counting order.

But this is precisely the number of divisors of $n$, since each way of representing $n = a b$ corresponds to the first term $a$ of the product.

Hence this sum is:


 * $\ds \sum_{n \mathop = 1}^\infty \frac {\map {\sigma_0} n} {z^n}$

as desired.