Dirichlet hyperbola method

Let $f$, $g$, and $h$ be multiplicative functions such that $f=g*h$, where $*$ denotes the convolution (http://planetmath.org/DirichletConvolution) of $g$ and $h$. The Dirichlet hyperbola method (typically shortened to hyperbola method) is a way to ${\displaystyle \sum _{n\le x}}f(n)$ by using the fact that $f=g*h$:

$$\sum _{n\le x}f(n)=\sum _{n\le x}\sum _{ab=n}g(a)h(b)=\sum _{a\le \sqrt{x}}\sum _{b\le \frac{x}{a}}g(a)h(b)+\sum _{b\le \sqrt{x}}\sum _{a\le \frac{x}{b}}g(a)h(b)-\sum _{a\le \sqrt{x}}\sum _{b\le \sqrt{x}}g(a)h(b)$$

Note that, since $ab=n\le x$, not both of $a$ and $b$ can be larger than $\sqrt{x}$. The Dirichlet hyperbola method follows from this fact as well as the inclusion-exclusion principle.

This method for calculating ${\displaystyle \sum _{n\le x}}f(n)$ is advantageous when the sums in of $g$ and $h$ are easier to handle and when $|g(n)-h(n)|$ is relatively small for most $n\in \mathbb{N}$.

As an example, the sum ${\displaystyle \sum _{n\le x}}\tau (n)$ will be calculated using the Dirichlet hyperbola method.

Note that $\tau =1*1$. Thus: