Square of Riemann Zeta Function

Theorem

Let \(d\) be the divisor counting function and \(\zeta\) the Riemann zeta function.

For \(\mathrm{Re}(s) > 1\)

\[ \zeta(s)^2 = \sum_{n = 1}^\infty \frac{d(n)}{n^s}.\]

Proof

This follows directly from the product of Dirichlet series using the Dirichlet series definition of \(\zeta\) and the fact that \(d = 1 \ast 1\).