Calculus + Number Theory = Calcumber Theory

For functions $f,g: \mathbb{N} \to \mathbb{C}$ (ie arithmetic functions) define the Dirichlet convolution $*$ by

$(f*g)(n) = \sum_{d|n} f(d)g(\frac{n}{d})$

It can be proven that

$\sum_{n=1}^\infty \frac{(f*g)(n)}{n^s} = \left[ \sum_{n=1}^\infty \frac{f(n)}{n^s} \right] \left[ \sum_{n=1}^\infty \frac{g(n)}{n^s} \right]$

It is clear to see that $\displaystyle \tau(n) = \sum_{d|n} 1 = (1*1)(n)$ , so

$\sum_{n=1}^\infty \frac{\tau(n)}{n^s} = \zeta^2(s)$

Applying this to $s = 4$ ,

$\sum_{n=1}^\infty \frac{\tau(n)}{n^s} = \boxed{\zeta^2(4)}$