# Square of Riemann Zeta Function

## Theorem

$\displaystyle \map {\zeta^2} z = \sum_{k \mathop = 1}^\infty \frac {\map d k} {k^z}$

where:

$\zeta$ is the Riemann zeta function
$d$ is the divisor function.

## Proof

 $\displaystyle \map {\zeta^2} z$ $=$ $\displaystyle \paren {\sum_{n \mathop = 1}^\infty \frac 1 {n^z} } \paren {\sum_{n \mathop = 1}^\infty \frac 1 {n^z} }$ $\displaystyle$ $=$ $\displaystyle \paren {1 + \frac 1 {2^z} + \frac 1 {3^z} + \frac 1 {4^z} + \frac 1 {5^z} + \frac 1 {6^z} + \cdots} \paren {1 + \frac 1 {2^z} + \frac 1 {3^z} + \frac 1 {4^z} + \frac 1 {5^z} + \frac 1 {6^z} + \cdots}$

Expanding this product, we get:

 $\displaystyle$ $=$ $\displaystyle 1 + \frac 1 {2^z} + \frac 1 {3^z} + \frac 1 {4^z} + \frac 1 {5^z} + \frac 1 {6^z} + \cdots$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \frac 1 {2^z} + \frac 1 {4^z} + \frac 1 {6^z} + \frac 1 {8^z} + \frac 1 {10^z} + \frac 1 {12^z} + \cdots$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \frac 1 {3^z} + \frac 1 {6^z} + \frac 1 {9^z} + \frac 1 {12^z} + \frac 1 {15^z} + \frac 1 {18^z} + \cdots$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \frac 1 {4^z} + \frac 1 {8^z} + \frac 1 {12^z} + \frac 1 {16^z} + \frac 1 {20^z} + \frac 1 {24^z} + \cdots$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \frac 1 {5^z} + \frac 1 {10^z} + \frac 1 {15^z} + \frac 1 {20^z} + \frac 1 {25^z} + \frac 1 {30^z} + \cdots$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \frac 1 {6^z} + \frac 1 {12^z} + \frac 1 {18^z} + \frac 1 {24^z} + \frac 1 {30^z} + \frac 1 {36^z} + \cdots$ $\displaystyle$  $\, \displaystyle \vdots \,$ $\displaystyle$ $\displaystyle$ $=$ $\displaystyle 1 + \frac 2 {2^z} + \frac 2 {3^z} + \frac 3 {4^z} + \frac 2 {5^z} + \frac 4 {6^z} + \cdots$

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:

$\displaystyle \sum_{n \mathop = 1}^\infty \frac {\map d n} {z^n}$

as desired.

$\blacksquare$