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:

Title	Dirichlet hyperbola method
Canonical name	DirichletHyperbolaMethod
Date of creation	2013-03-22 15:58:27
Last modified on	2013-03-22 15:58:27
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	14
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 11N37
Synonym	hyperbola method
Related topic	ConvolutionMethod
Related topic	TauFunction
Related topic	EulersConstant