Editing Inclusion–exclusion principle (section)

===Dirichlet hyperbola method===
{{main|Dirichlet hyperbola method}}

[[File:Dirichlet hyperbola example_4.svg|thumb|An example of the Dirichlet hyperbola method with <math>n = 10,</math> <math>a \approx 2.7,</math> and <math>b \approx 3.7.</math>]]

The Dirichlet hyperbola method re-expresses a sum of a [[multiplicative function]] <math>f(n)</math> by selecting a suitable [[Dirichlet convolution]] <math>f = g \ast h</math>, recognizing that the sum

: <math>F(n) = \sum_{k=1}^n f(k) = \sum_{k=1}^n \sum_{xy=k}^{} g(x) h(y)</math>

can be recast as a sum over the [[lattice points]] in a region bounded by <math>x \geq 1</math>, <math>y \geq 1</math>, and <math>xy \leq n</math>, splitting this region into two overlapping subregions, and finally using the inclusion–exclusion principle to conclude that

: <math>F(n) = \sum_{k=1}^n f(k)
= \sum_{k=1}^n \sum_{xy=k}^{} g(x)h(y)
= \sum_{x=1}^a \sum_{y=1}^{n/x} g(x)h(y) + \sum_{y=1}^b \sum_{x=1}^{n/y} g(x)h(y) - \sum_{x=1}^a \sum_{y=1}^b g(x)h(y).</math>