Square of Riemann Zeta Function

Theorem

 * $\displaystyle \zeta^2 \left({z}\right) = \sum_{k \mathop = 1}^\infty \frac{d \left({k}\right) } {k^z}$

where:
 * $\zeta$ is the Riemann zeta function
 * $d$ is the divisor 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 $ab$, counting order.

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

Hence this sum is:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac {d \left({n}\right) } {z^n}$

as desired.