Dirichlet hyperbola method

Let $f$ , $g$ , and $h$ be multiplicative functions such that $f=g*h$ , where $*$ denotes the convolution of $g$ and $h$ . The Dirichlet hyperbola method (typically shortened to hyperbola method) is a way to calculate $\displaystyle \sum_{n \le x} f(n)$ 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 terms 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: